Chebyshev polynomials: Difference between revisions

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In combinatorics, Ramsey's theorem states that in any colouring of the edges of a sufficiently large complete graph, one will find monochromatic complete subgraphs. For two colours, Ramsey's theorem states that for any pair of positive integers (r,s), there exists a least positive integer R(r,s) such that for any complete graph on R(r,s) vertices, whose edges are coloured red or blue, there exists either a complete subgraph on r vertices which is entirely blue, or a complete subgraph on s vertices which is entirely red. Here R(r,s) signifies an integer that depends on both r and s. It is understood to represent the smallest integer for which the theorem holds.

Ramsey's theorem is a foundational result in combinatorics. The first version of this result was proved by F. P. Ramsey. This initiated the combinatorial theory, now called Ramsey theory, that seeks regularity amid disorder: general conditions for the existence of substructures with regular properties. In this application it is a question of the existence of monochromatic subsets, that is, subsets of connected edges of just one colour.

An extension of this theorem applies to any finite number of colours, rather than just two. More precisely, the theorem states that for any given number of colours c, and any given integers n₁,...,n_c, there is a number, R(n₁, ..., n_c), such that if the edges of a complete graph of order R(n₁, ..., n_c) are coloured with c different colours, then for some i between 1 and c, it must contain a complete subgraph of order n_i whose edges are all colour i. The special case above has c = 2 (and n₁ = r and n₂ = s).

Example: R(3,3) = 6

In the following example, the formula R(3,3) provides a solution to the question which asks the minimum number of vertices a graph must contain in order to ensure that either (1) at least 3 vertices in the graph are connected or (2) at least 3 vertices in the graph are unconnected. Note that owing to the symmetrical nature of the problem space, R(r,s) is equal to R(s,r).

Suppose the edges of a complete graph on 6 vertices are coloured red and blue. Pick a vertex v. There are 5 edges incident to v and so (by the pigeonhole principle) at least 3 of them must be the same colour. Without loss of generality we can assume at least 3 of these edges, connecting to vertices r, s and t, are blue. (If not, exchange red and blue in what follows.) If any of the edges (r, s), (r, t), (s, t) are also blue then we have an entirely blue triangle. If not, then those three edges are all red and we have an entirely red triangle. Since this argument works for any colouring, any K₆ contains a monochromatic K₃, and therefore R(3,3) ≤ 6. The popular version of this is called the theorem on friends and strangers.

An alternative proof works by double counting. It goes as follows: Count the number of ordered triples of vertices x, y, z such that the edge (xy) is red and the edge (yz) is blue. Firstly, any given vertex will be the middle of either 0 × 5 = 0 (all edges from the vertex are the same colour), 1 × 4 = 4 (four are the same colour, one is the other colour), or 2 × 3 = 6 (three are the same colour, two are the other colour) such triples. Therefore there are at most 6 × 6 = 36 such triples. Secondly, for any non-monochromatic triangle (xyz), there exist precisely two such triples. Therefore there are at most 18 non-monochromatic triangles. Therefore at least 2 of the 20 triangles in the K₆ are monochromatic.

Conversely, it is possible to 2-colour a K₅ without creating any monochromatic K₃, showing that R(3,3) > 5. The unique colouring is shown to the right. Thus R(3,3) = 6.

The task of proving that R(3,3) ≤ 6 was one of the problems of William Lowell Putnam Mathematical Competition in 1953.

Proof of the theorem

2-colour case

First we prove the theorem for the 2-colour case, by induction on r + s. It is clear from the definition that for all n, R(n, 1) = R(1, n) = 1. This starts the induction. We prove that R(r, s) exists by finding an explicit bound for it. By the inductive hypothesis R(r − 1, s) and R(r, s − 1) exist.

Lemma 1. R(r, s) ≤ R(r − 1, s) + R(r, s − 1):

Proof. Consider a complete graph on R(r − 1, s) + R(r, s − 1) vertices whose edges are coloured with two colours. Pick a vertex v from the graph, and partition the remaining vertices into two sets M and N, such that for every vertex w, w is in M if (v, w) is blue, and w is in N if (v, w) is red. Because the graph has R(r − 1, s) + R(r, s − 1) = |M| + |N| + 1 vertices, it follows that either |M| ≥ R(r − 1, s) or |N| ≥ R(r, s − 1). In the former case, if M has a red K_s then so does the original graph and we are finished. Otherwise M has a blue K_r−1 and so M ∪ {v} has blue K_r by definition of M. The latter case is analogous.Thus the claim is true and we have completed the proof for 2 colours.

Note. In the 2-colour case, if R(r − 1, s) and R(r, s − 1) are both even, the induction inequality can be strengthened to:^[1]

R(r, s) ≤ R(r − 1, s) + R(r, s − 1) − 1.

General case

We now prove the result for the general case of c colours. The proof is again by induction, this time on the number of colours c. We have the result for c = 1 (trivially) and for c = 2 (above). Now let c > 2.

Lemma 2. R(n₁, ..., n_c) ≤ R(n₁, ..., n_c−2, R(n_c−1, n_c)).

Proof. Consider a graph on t vertices and colour its edges with c colours. Now 'go colour-blind' and pretend that c − 1 and c are the same colour. Thus the graph is now (c − 1)-coloured. By the inductive hypothesis, it contains either a K_n_{_i} monochromatically coloured with colour i for some 1 ≤ i ≤ c − 2 or a K_R(n_{_c−1,n_c)}-coloured in the 'blurred colour'. In the former case we are finished. In the latter case, we recover our sight again and see from the definition of R(n_c−1, n_c) we must have either a (c − 1)-monochrome K_n_{_c−1} or a c-monochrome K_n_{_c}. In either case the proof is complete.

The right hand side of the inequality in Lemma 2 only contains Ramsey numbers for c − 1 colours and 2 colours, and therefore exists and is a finite number t, by the inductive hypothesis. Thus, proving the claim will prove the theorem.

Ramsey numbers

The numbers R(r,s) in Ramsey's theorem (and their extensions to more than two colours) are known as Ramsey numbers. An upper bound for R(r,s) can be extracted from the proof of the theorem, and other arguments give lower bounds. (The first lower bound was obtained by Paul Erdős using the probabilistic method.) However, there is a vast gap between the tightest lower bounds and the tightest upper bounds. Consequently, there are very few numbers r and s for which we know the exact value of R(r,s). Computing a lower bound L for R(r,s) usually requires exhibiting a blue/red colouring of the graph K_L−1 with no blue K_r subgraph and no red K_s subgraph. Upper bounds are often considerably more difficult to establish: one either has to check all possible colourings to confirm the absence of a counterexample, or to present a mathematical argument for its absence. A sophisticated computer program does not need to look at all colourings individually in order to eliminate all of them; nevertheless it is a very difficult computational task that existing software can only manage on small sizes. The complexity for searching all possible graphs (via brute force) is O(2^{(n−1)(n−2)/2}) for an upper bound of n nodes.^[2]

As described above, R(3,3) = 6. It is easy to prove that R(4,2) = 4, and, more generally, that R(s,2) = s for all s: a graph on s − 1 nodes with all edges coloured red serves as a counterexample and proves that R(s,2) ≥ s ; among colourings of a graph on s nodes, the colouring with all edges coloured red contains a s-node red subgraph, and all other colourings contain a 2-node blue subgraph (that is, a pair of nodes connected with a blue edge.) Using induction inequalities, it can be concluded that R(4,3) ≤ R(4,2) + R(3,3) − 1 = 9, and therefore R(4,4) ≤ R(4,3) + R(3,4) ≤ 18. There are only two (4,4,16) graphs (that is, 2-colourings of a complete graph on 16 nodes without 4-node red or blue complete subgraphs) among 6.4×10²² different 2-colourings of 16-node graphs, and only one (4,4,17) graph (the Paley graph of order 17) among 2.46×10²⁶ colourings.^[3] (This was proven by Evans, Pulham and Sheehan in 1979.) It follows that R(4,4) = 18.

The fact that R(4,5)=25 was first established by Brendan McKay and Stanisław Radziszowski in 1995.^[4]

The exact value of R(5,5) is unknown, although it is known to lie between 43 (Geoffrey Exoo) and 49 (McKay and Radziszowski) (inclusive).

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McKay, Radziszowski and Exoo employed computer-assisted graph generation methods to conjecture in 1997 that R(5,5) is exactly 43. They were able to construct exactly 656 (5,5,42) graphs, arriving at the same set of graphs through different routes. None of the 656 graphs can be extended to a (5,5,43) graph.^[5]

For R(r,s) with r, s > 5, only weak bounds are available. Lower bounds for R(6,6) and R(8,8) have not been improved since 1965 and 1972, respectively.^[6]

R(r,s) for values of r and s up to 10 are shown in the table below. Where the exact value is unknown, the table lists the best known bounds. R(r,s) for values of r and s less than 3 are given by R(1,s) = 1 and R(2,s) = s for all values of s. The standard survey on the development of Ramsey number research has been written by Stanisław Radziszowski.

r,s	1	2	3	4	5	6	7	8	9	10
1	1
2	1	2
3	1	3	6
4	1	4	9	18
5	1	5	14	25	43–49
6	1	6	18	36–41	58–87	102–165
7	1	7	23	49–61	80–143	113–298	205–540
8	1	8	28	58–84	101–216	132–495	217–1031	282–1870
9	1	9	36	73–115	126–316	169–780	241–1713	317–3583	565–6588
10	1	10	40–42	92–149	144–442	179–1171	289–2826	331–6090	581–12677	798–23556

Since R(r, s) = R(s, r), there is a trivial symmetry across the diagonal.

This table is extracted from a larger table compiled by Stanisław Radziszowski.^[6]

Asymptotics

The inequality R(r, s) ≤ R(r − 1, s) + R(r, s − 1) may be applied inductively to prove that

R(r,s)\leq {\binom {r+s-2}{r-1}}.

In particular, this result, due to Erdős and Szekeres, implies that when r = s,

R(s,s)\leq (1+o(1)){\frac {4^{s-1}}{\sqrt {\pi s}}}.

An exponential lower bound,

R(s,s)\geq (1+o(1)){\frac {s}{{\sqrt {2}}e}}2^{\frac {s}{2}},

was given by Erdős in 1947 and was instrumental in his introduction of the probabilistic method. There is obviously a huge gap between these two bounds: for example, for s = 10, this gives 101 ≤ R(10,10) ≤ 48620. Nevertheless, exponential growth factors of either bound have not been improved to date and still stand at 4 and ${\sqrt {2}}$ respectively. There is no known explicit construction producing an exponential lower bound. The best known lower and upper bounds for diagonal Ramsey numbers currently stand at

(1+o(1)){\frac {{\sqrt {2}}s}{e}}2^{\frac {s}{2}}\leq R(s,s)\leq s^{-{\frac {c\log s}{\log \log s}}}4^{s},

due to Spencer and Conlon respectively.

For the off-diagonal Ramsey numbers R(3,t), it is known that they are of order ${\tfrac {t^{2}}{\log t}}$ ; this may be stated equivalently as saying that the smallest possible independence number in an n-vertex triangle-free graph is $\Theta ({\sqrt {n\log n}})$ . The upper bound for R(3,t) is given by Ajtai, Komlós, and Szemerédi, the lower bound by Kim. More generally, for off-diagonal Ramsey numbers R(s, t) with s fixed and t growing, the best known bounds are

c'_{s}{\frac {t^{\frac {s+1}{2}}}{(\log t)^{{\frac {s+1}{2}}-{\frac {1}{s-2}}}}}\leq R(s,t)\leq c_{s}{\frac {t^{s-1}}{(\log t)^{s-2}}},

due to Bohman and Keevash and Ajtai, Komlós and Szemerédi respectively.

A multicolour example: R(3,3,3) = 17

The only two 3-colourings of K₁₆ with no monochromatic K₃. The untwisted colouring (top) and the twisted colouring (bottom).

A multicolour Ramsey number is a Ramsey number using 3 or more colours. There is only one nontrivial multicolour Ramsey number for which the exact value is known, namely R(3,3,3) = 17.

Suppose that you have an edge colouring of a complete graph using 3 colours, red, yellow and green. Suppose further that the edge colouring has no monochromatic triangles. Select a vertex v. Consider the set of vertices that have a green edge to the vertex v. This is called the green neighborhood of v. The green neighborhood of v cannot contain any green edges, since otherwise there would be a green triangle consisting of the two endpoints of that green edge and the vertex v. Thus, the induced edge colouring on the green neighborhood of v has edges coloured with only two colours, namely yellow and red. Since R(3,3) = 6, the green neighborhood of v can contain at most 5 vertices. Similarly, the red and yellow neighborhoods of v can contain at most 5 vertices each. Since every vertex, except for v itself, is in one of the green, red or yellow neighborhoods of v, the entire complete graph can have at most 1 + 5 + 5 + 5 = 16 vertices. Thus, we have R(3,3,3) ≤ 17.

To see that R(3,3,3) ≥ 17, it suffices to draw an edge colouring on the complete graph on 16 vertices with 3 colours that avoids monochromatic triangles. It turns out that there are exactly two such colourings on K₁₆, the so-called untwisted and twisted colourings. Both colourings are shown in the figure to the right, with the untwisted colouring on the top, and the twisted colouring on the bottom. In both colourings in the figure, note that the vertices are labeled, and that the vertices v₁₁ through v₁₅ are drawn twice, on both the left and the right, in order to simplify the drawings.

Thus, R(3,3,3) = 17.

If you select any colour of either the untwisted or twisted colouring on K₁₆, and consider the graph whose edges are precisely those edges that have the specified colour, you will get the Clebsch graph.

It is known that there are exactly two edge colourings with 3 colours on K₁₅ that avoid monochromatic triangles, which can be constructed by deleting any vertex from the untwisted and twisted colourings on K₁₆, respectively.

It is also known that there are exactly 115 edge colourings with 3 colours on K₁₄ that avoid monochromatic triangles, provided that we consider edge colourings that differ by a permutation of the colours as being the same.

Extensions of the theorem

The theorem can also be extended to hypergraphs. An m-hypergraph is a graph whose "edges" are sets of m vertices – in a normal graph an edge is a set of 2 vertices. The full statement of Ramsey's theorem for hypergraphs is that for any integers m and c, and any integers n₁,...,n_c, there is an integer R(n₁, ..., n_c;c,m) such that if the hyperedges of a complete m-hypergraph of order R(n₁,...,n_c;c,m) are coloured with c different colours, then for some i between 1 and c, the hypergraph must contain a complete sub-m-hypergraph of order n_i whose hyperedges are all colour i. This theorem is usually proved by induction on m, the 'hyper-ness' of the graph. The base case for the proof is m = 2, which is exactly the theorem above.

Infinite Ramsey theorem

A further result, also commonly called Ramsey's theorem, applies to infinite graphs. In a context where finite graphs are also being discussed it is often called the "Infinite Ramsey theorem". As intuition provided by the pictorial representation of a graph is diminished when moving from finite to infinite graphs, theorems in this area are usually phrased in set-theoretic terminology.

Theorem. Let X be some countably infinite set and colour the elements of X⁽ⁿ⁾ (the subsets of X of size n) in c different colours. Then there exists some infinite subset M of X such that the size n subsets of M all have the same colour.

Proof: The proof is given for c = 2. It is easy to prove the theorem for an arbitrary number of colours using a 'colour-blindness' argument as above. The proof is by (complete) induction on n, the size of the subsets. For n = 1,the statement is equivalent to saying that if you split an infinite set into two sets, one of them is infinite. This is evident. Assuming the theorem is true for n ≤ r, we prove it for n = r + 1. Given a 2-colouring of the (r + 1)-element subsets of X, let a₀ be an element of X and let Y = X\{a₀}. We then induce a 2-colouring of the r-element subsets of Y, by just adding a₀ to each r-element subset (to get an (r + 1)-element subset of X). By the induction hypothesis, there exists an infinite subset Y₁ within Y such that every r-element subset of Y₁ is coloured the same colour in the induced colouring. Thus there is an element a₀ and an infinite subset Y₁ such that all the (r + 1)-element subsets of X consisting of a₀ and r elements of Y₁ have the same colour. By the same argument, there is an element a₁ in Y₁ and an infinite subset Y₂ of Y₁ with the same properties. Inductively, we obtain a sequence {a₀,a₁,a₂,...} such that the colour of each (r + 1)-element subset (a_i(1),a_i(2),...,a_i(r + 1)) with i(1) < i(2) < ... < i(r + 1) depends only on the value of i(1). Further, there are infinitely many values of i(n) such that this colour will be the same. Take these a_i(n)'s to get the desired monochromatic set.

Infinite version implies the finite

It is possible to deduce the finite Ramsey theorem from the infinite version by a proof by contradiction. Suppose the finite Ramsey theorem is false. Then there exist integers c, n, T such that for every integer k, there exists a c-colouring of $[k]^{(n)}$ without a monochromatic set of size T. Let C_k denote the c-colourings of $[k]^{(n)}$ without a monochromatic set of size T.

For any k, the restriction of a colouring in C_k+1 to $[k]^{(n)}$ (by ignoring the colour of all sets containing k+1) is a colouring in C_k. Define $C_{k}^{1}$ to be the colourings in C_k which are restrictions of colourings in C_k+1. Since C_k+1 is not empty, neither is $C_{k}^{1}$ .

Similarly, the restriction of any colouring in $C_{k+1}^{1}$ is in $C_{k}^{1}$ , allowing one to define $C_{k}^{2}$ as the set of all such restrictions, a non-empty set. Continuing so, define $C_{k}^{m}$ for all integers m, k.

Now, for any integer k, $C_{k}\supseteq C_{k}^{1}\supseteq C_{k}^{2}\supseteq \dots$ , and each set is non-empty. Furthermore, C_k is finite as $|C_{k}|\leq c^{\frac {k!}{n!(k-n)!}}$ . It follows that the intersection of all of these sets is non-empty, and let $D_{k}=C_{k}\cap C_{k}^{1}\cap C_{k}^{2}\cap \dots$ . Then every colouring in D_k is the restriction of a colouring in D_k+1. Therefore, by unrestricting a colouring in D_k to a colouring in D_k+1, and continuing doing so, one constructs a colouring of $\mathbb {N} ^{(n)}$ without any monochromatic set of size T. This contradicts the infinite Ramsey theorem.

If a suitable topological viewpoint is taken, this argument becomes a standard compactness argument showing that the infinite version of the theorem implies the finite version.

Directed graph Ramsey numbers

It is also possible to define Ramsey numbers for directed graphs. (These were introduced by P. Erdős & L. Moser.) Let R(n) be the smallest number Q such that any complete graph with singly directed arcs (also called a "tournament") and with ≥ Q nodes contains an acyclic (also called "transitive") n-node subtournament.

This is the directed-graph analogue of what (above) has been called R(n,n;2), the smallest number Z such that any 2-colouring of the edges of a complete undirected graph with ≥ Z nodes, contains a monochromatic complete graph on n nodes. (The directed analogue of the two possible arc colours is the two directions of the arcs, the analogue of "monochromatic" is "all arc-arrows point the same way," i.e. "acyclic.")

Indeed many Template:Who find the directed graph problem to actually be more elegant than the undirected one. We have R(0)=0, R(1)=1, R(2)=2, R(3)=4, R(4)=8, R(5)=14, R(6)=28, 32≤R(7)≤55, and R(8) is again a problem you do not want powerful aliens to pose.

Notes

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References

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Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.

External links

DTZ's auction group in Singapore auctions all types of residential, workplace and retail properties, retailers, homes, accommodations, boarding houses, industrial buildings and development websites. Auctions are at the moment held as soon as a month.

Whitehaven @ Pasir Panjang – A boutique improvement nicely nestled peacefully in serene Pasir Panjang personal estate presenting a hundred and twenty rare freehold private apartments tastefully designed by the famend Ong & Ong Architect. Only a short drive away from Science Park and NUS Campus, Jade Residences, a recent Freehold condominium which offers high quality lifestyle with wonderful facilities and conveniences proper at its door steps. Its fashionable linear architectural fashion promotes peace and tranquility living nestled within the D19 personal housing enclave. Rising workplace sector leads real estate market efficiency, while prime retail and enterprise park segments moderate and residential sector continues in decline International Market Perspectives - 1st Quarter 2014

There are a lot of websites out there stating to be one of the best seek for propertycondominiumhouse, and likewise some ways to discover a low cost propertycondominiumhouse. Owning a propertycondominiumhouse in Singapore is the dream of virtually all individuals in Singapore, It is likely one of the large choice we make in a lifetime. Even if you happen to're new to Property listing singapore funding, we are right here that will help you in making the best resolution to purchase a propertycondominiumhouse at the least expensive value.

Jun 18 ROCHESTER in MIXED USE IMPROVEMENT $1338000 / 1br - 861ft² - (THE ROCHESTER CLOSE TO NORTH BUONA VISTA RD) pic real property - by broker Jun 18 MIXED USE IMPROVEMENT @ ROCHESTER @ ROCHESTER PK $1880000 / 1br - 1281ft² - (ROCHESTER CLOSE TO NORTH BUONA VISTA) pic real estate - by broker Tue 17 Jun Jun 17 Sunny Artwork Deco Gem Near Seashore-Super Deal!!! $103600 / 2br - 980ft² - (Ventnor) pic actual estate - by owner Jun 17 Freehold semi-d for rent (Jalan Rebana ) $7000000 / 5909ft² - (Jalan Rebana ) actual property - by dealer Jun sixteen Ascent @ 456 in D12 (456 Balestier Highway,Singapore) pic real property - by proprietor Jun 16 RETAIL SHOP AT SIM LIM SQUARE FOR SALE, IT MALL, ROCHOR, BUGIS MRT $2000000 / 506ft² - (ROCHOR, BUGIS MRT) pic real estate - by dealer HDB Scheme Any DBSS BTO

In case you are eligible to purchase landed houses (open solely to Singapore residents) it is without doubt one of the best property investment choices. Landed housing varieties solely a small fraction of available residential property in Singapore, due to shortage of land right here. In the long term it should hold its worth and appreciate as the supply is small. In truth, landed housing costs have risen the most, having doubled within the last eight years or so. However he got here back the following day with two suitcases full of money. Typically we've got to clarify to such folks that there are rules and paperwork in Singapore and you can't just buy a home like that,' she said. For conveyancing matters there shall be a recommendedLondon Regulation agency familiar with Singapore London propertyinvestors to symbolize you

Sales transaction volumes have been expected to hit four,000 units for 2012, close to the mixed EC gross sales volume in 2010 and 2011, in accordance with Savills Singapore. Nevertheless the last quarter was weak. In Q4 2012, sales transactions were 22.8% down q-q to 7,931 units, in line with the URA. The quarterly sales discount was felt throughout the board. When the sale just starts, I am not in a hurry to buy. It's completely different from a private sale open for privileged clients for one day solely. Orchard / Holland (D09-10) House For Sale The Tembusu is a singular large freehold land outdoors the central area. Designed by multiple award-profitable architects Arc Studio Architecture + Urbanism, the event is targeted for launch in mid 2013. Post your Property Condos Close to MRT

Other Sports Official Kull from Drumheller, has hobbies such as telescopes, property developers in singapore and crocheting. Identified some interesting places having spent 4 months at Saloum Delta.

my web-site http://himerka.com/
Ramsey@Home is a distributed computing project designed to find new lower bounds for various Ramsey numbers using a host of different techniques.
Radziszowski's survey of small Ramsey numbers
this lisp code computes upper bounds for particular Ramsey numbers given the general rules in Radziszowski
Survey of directed-graph Ramsey numbers
Ramsey Number – from MathWorld (contains lower and upper bounds up to R(19,19))
Ramsey Number – Geoffrey Exoo (Contains R(5,5) > 42 counter-proof)
A proof that R(3,6) = 18

↑ Template:Cite web
↑ 2.6 Ramsey Theory from Mathematics Illuminated
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↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
↑ ^6.0 ^6.1 Template:Cite web

[1] Template:Cite web

[2] 2.6 Ramsey Theory from Mathematics Illuminated

[3] Template:Cite web

[4] One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

[5] One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

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Chebyshev polynomials: Difference between revisions

Revision as of 11:46, 29 January 2014

Contents

Example: R(3,3) = 6

Proof of the theorem

2-colour case

General case

Ramsey numbers

Asymptotics

A multicolour example: R(3,3,3) = 17

Extensions of the theorem

Infinite Ramsey theorem

Infinite version implies the finite

Directed graph Ramsey numbers

See also

Notes

References

External links

Navigation menu

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-Let's look an actual registry scan plus a few of what you will see whenever we do 1 on the computer. This test was done on a computer which was not functioning as it could, operating at slow speed plus having several issues with freezing up.<br><br>You need to recognize several simple and cheap ways that will solve the problem of your computer and speed it up. The earlier we fix it, the less damage a computer gets. I can tell regarding some practical techniques that may assist we to accelerate we computer.<br><br>So, this advanced dual scan is not only among the greater, yet it is equally freeware. And as of all of this which countless respect CCleaner among the better registry cleaners inside the market now. I would add which I personally choose Regcure for the simple reason that it has a greater interface and I understand for a truth it is ad-ware without charge.<br><br>It is general that the imm32.dll error is caused due to a mis-deletion activity. If you cannot find the imm32.dll anywhere on a computer, there is not any doubt which it should be mis-deleted whenever uninstalling programs or other unneeded files. Hence, you are able to straight cope it from additional programs or download it from a secure internet and then put it on a computer.<br><br>One other way whenever arresting the 1328 error is to wash out your PC's registry. The registry is surprisingly important because it really is where settings and files used by Windows for running are stored. As it really is frequently selected, breakdowns and situations of files getting corrupted are not uncommon. Additionally because of the means it is very configured, the "registry" gets saved inside the incorrect fashion constantly, which makes your program run slow, eventually causing a PC to suffer from a series of mistakes. The most effective system 1 may use inside cleaning out registries is to use a reliable [http://bestregistrycleanerfix.com/tune-up-utilities tuneup utilities] program. A registry cleaner can seek out and repair corrupted registry files plus settings permitting one's computer to run usually again.<br><br>Active X controls are used over the entire spectrum of computer and internet technologies. These controls are referred to as the building blocks of the internet plus as the glue which puts it all together. It is a standard that is selected by all programmers to create the internet more useful and interactive. Without these control specifications there would basically be no public web.<br><br>Across the best of the scan results display page you see the tabs... Registry, Junk Files, Privacy, Bad Active X, Performance, etc. Each of these tabs may show we the results of that region. The Junk Files are mostly temporary files such as internet information, pictures, internet pages... And they are simply taking up storage space.<br><br>There are many companies which offer the service of troubleshooting your PC every time we call them, all you have to do is sign up with them plus for a tiny fee, you have your machine always functioning effectively and serve you greater.
+{{See introduction|Ramsey theory}}
+In [[combinatorics]], '''Ramsey's theorem''' states that in any colouring of the [[edge (graph theory)|edges]] of a sufficiently large [[complete graph]], one will find monochromatic complete subgraphs. For two colours, Ramsey's theorem states that for any pair of positive integers (''r'',''s''), there exists a least positive integer ''R''(''r'',''s'') such that for any [[complete graph]] on ''R''(''r'',''s'') vertices, whose edges are coloured ''red'' or ''blue'', there exists either a complete subgraph on ''r'' vertices which is entirely blue, or a complete subgraph on ''s'' vertices which is entirely red. Here ''R''(''r'',''s'') signifies an integer that depends on both ''r'' and ''s''. It is understood to represent the smallest integer for which the theorem holds.
+Ramsey's theorem is a foundational result in combinatorics. The first version of this result was proved by [[F. P. Ramsey]]. This initiated the combinatorial theory, now called [[Ramsey theory]], that seeks regularity amid disorder: general conditions for the existence of substructures with regular properties. In this application it is a question of the existence of '''''monochromatic subsets''''', that is, subsets of connected edges of just one colour.
+An extension of this theorem applies to any finite number of colours, rather than just two. More precisely, the theorem states that for any given number of colours ''c'', and any given integers ''n''<sub>1</sub>,...,''n<sub>c</sub>'', there is a number, ''R''(''n''<sub>1</sub>, ..., ''n<sub>c</sub>''), such that if the edges of a complete graph of
+order ''R''(''n''<sub>1</sub>, ..., ''n''<sub>''c''</sub>) are coloured with ''c'' different colours, then for some ''i'' between 1 and ''c'', it must contain a complete subgraph of order ''n''<sub>''i''</sub> whose edges are all colour ''i''. The special case above has ''c'' = 2 (and ''n''<sub>1</sub> = ''r'' and ''n''<sub>2</sub> = ''s'').
+==Example: ''R''(3,3) = 6==
+[[Image:RamseyTheory K5 no mono K3.svg|frame|A 2-colouring of ''K''<sub>5</sub> with no monochromatic ''K''<sub>3</sub>]]
+In the following example, the formula ''R''(3,3) provides a solution to the question which asks the minimum number of vertices a graph must contain in order to ensure that either (1) at least 3 vertices in the graph are connected '''or''' (2) at least 3 vertices in the graph are unconnected. Note that owing to the symmetrical nature of the problem space, ''R''(''r'',''s'') is equal to ''R''(''s'',''r'').
+Suppose the edges of a complete graph on 6 vertices are coloured red and blue. Pick a vertex ''v''. There are 5 edges incident to ''v'' and so (by the [[pigeonhole principle]]) at least 3 of them must be the same colour. [[Without loss of generality]] we can assume at least 3 of these edges, connecting to vertices ''r'', ''s'' and ''t'', are blue. (If not, exchange red and blue in what follows.) If any of the edges (''r'', ''s''), (''r'', ''t''), (''s'', ''t'') are also blue then we have an entirely blue triangle. If not, then those three edges are all red and we have an entirely red triangle. Since this argument works for any colouring, ''any'' ''K''<sub>6</sub> contains a monochromatic ''K''<sub>3</sub>, and therefore ''R''(3,3) ≤ 6. The popular version of this is called the [[theorem on friends and strangers]].
+An alternative proof works by [[double counting (proof technique)|double counting]]. It goes as follows: Count the number of ordered triples of vertices ''x'', ''y'', ''z'' such that the edge (''xy'') is red and the edge (''yz'') is blue. Firstly, any given vertex will be the middle of either 0&nbsp;×&nbsp;5&nbsp;=&nbsp;0 (all edges from the vertex are the same colour), 1&nbsp;×&nbsp;4&nbsp;=&nbsp;4 (four are the same colour, one is the other colour), or 2&nbsp;×&nbsp;3&nbsp;=&nbsp;6 (three are the same colour, two are the other colour) such triples. Therefore there are at most 6&nbsp;×&nbsp;6&nbsp;=&nbsp;36 such triples. Secondly, for any non-monochromatic triangle ('''xyz'''), there exist precisely two such triples. Therefore there are at most 18 non-monochromatic triangles. Therefore at least 2 of the 20 triangles in the ''K''<sub>6</sub> are monochromatic.
+Conversely, it is possible to 2-colour a ''K''<sub>5</sub> without creating any monochromatic ''K''<sub>3</sub>, showing that ''R''(3,3)&nbsp;>&nbsp;5. The unique colouring is shown to the right. Thus ''R''(3,3)&nbsp;=&nbsp;6.
+The task of proving that ''R''(3,3)&nbsp;≤&nbsp;6 was one of the problems of [[William Lowell Putnam Mathematical Competition]] in 1953.
+==Proof of the theorem==
+===2-colour case===
+First we prove the theorem for the 2-colour case, by [[Mathematical induction|induction]] on ''r''&nbsp;+&nbsp;''s''. It is clear from the definition that for all ''n'', ''R''(''n'', 1) = ''R''(1, ''n'') = 1. This starts the induction. We prove that ''R''(''r'', ''s'') exists by finding an explicit bound for it. By the inductive hypothesis ''R''(''r''&nbsp;−&nbsp;1, ''s'') and ''R''(''r'', ''s''&nbsp;−&nbsp;1) exist.
+<blockquote>'''Lemma 1.''' ''R''(''r'', ''s'') ≤ ''R''(''r''&nbsp;−&nbsp;1, ''s'') + ''R''(''r'', ''s''&nbsp;−&nbsp;1):</blockquote>
+'''Proof.''' Consider a complete graph on ''R''(''r''&nbsp;−&nbsp;1, ''s'') + ''R''(''r'', ''s''&nbsp;−&nbsp;1) vertices whose edges are coloured with two colours. Pick a vertex ''v'' from the graph, and partition the remaining vertices into two sets ''M'' and ''N'', such that for every vertex ''w'', ''w'' is in ''M'' if (''v'', ''w'') is blue, and ''w'' is in ''N'' if (''v'', ''w'') is red. Because the graph has ''R''(''r'' − 1, ''s'') + ''R''(''r'', ''s'' − 1) = |''M''| + |''N''| + 1 vertices, it follows that either |''M''| ≥ ''R''(''r'' − 1, ''s'') or |''N''| ≥ ''R''(''r'', ''s'' − 1). In the former case, if ''M'' has a red ''K<sub>s</sub>'' then so does the original graph and we are finished. Otherwise ''M'' has a blue ''K''<sub>''r''−1</sub> and so ''M'' ∪ {''v''} has blue ''K<sub>r</sub>'' by definition of ''M''. The latter case is analogous.Thus the claim is true and we have completed the proof for 2 colours.
+'''Note.''' In the 2-colour case, if ''R''(''r''&nbsp;−&nbsp;1, ''s'') and ''R''(''r'', ''s''&nbsp;−&nbsp;1) are both even, the induction inequality can be strengthened to:<ref>{{cite web|url=http://www.cut-the-knot.org/Curriculum/Combinatorics/ThreeOrThree.shtml#inequality2 |title=Party Acquaintances}}</ref>
+:''R''(''r'', ''s'') ≤ ''R''(''r''&nbsp;−&nbsp;1, ''s'') + ''R''(''r'', ''s''&nbsp;−&nbsp;1)&nbsp;−&nbsp;1.
+===General case===
+We now prove the result for the general case of ''c'' colours. The proof is again by induction, this time on the number of colours ''c''. We have the result for ''c'' = 1 (trivially) and for ''c'' = 2 (above). Now let ''c'' > 2.
+<blockquote>'''Lemma 2.''' ''R''(''n''<sub>1</sub>, ..., ''n''<sub>''c''</sub>) ≤ ''R''(''n''<sub>1</sub>, ..., ''n''<sub>''c''−2</sub>, ''R''(''n''<sub>''c''−1</sub>, ''n''<sub>''c''</sub>)).</blockquote>
+'''Proof.''' Consider a graph on ''t'' vertices and colour its edges with ''c'' colours. Now 'go colour-blind' and pretend that ''c''&nbsp;−&nbsp;1 and ''c'' are the same colour. Thus the graph is now (''c''&nbsp;−&nbsp;1)-coloured. By the inductive hypothesis, it contains either a ''K<sub>n</sub><sub><sub>i</sub></sub>'' monochromatically coloured with colour ''i'' for some 1 ≤ ''i'' ≤ ''c''&nbsp;−&nbsp;2 or a ''K''<sub>''R''(''n''</sub><sub><sub>''c''−1</sub>,''n''<sub>''c''</sub>)</sub>-coloured in the 'blurred colour'. In the former case we are finished. In the latter case, we recover our sight again and see from the definition of ''R''(''n<sub>c−1</sub>'', ''n<sub>c</sub>'') we must have either a (''c''&nbsp;−&nbsp;1)-monochrome ''K''<sub>''n''</sub><sub><sub>''c''−1</sub></sub> or a ''c''-monochrome ''K''<sub>''n''</sub><sub><sub>''c''</sub></sub>. In either case the proof is complete.
+The right hand side of the inequality in Lemma 2 only contains Ramsey numbers for ''c''&nbsp;−&nbsp;1 colours and 2 colours, and therefore exists and is a finite number ''t'', by the inductive hypothesis. Thus, proving the claim will prove the theorem.
+==Ramsey numbers==<!-- This section is linked from [[Unsolved problems in mathematics]] -->
+The numbers ''R''(''r'',''s'') in Ramsey's theorem (and their extensions to more than two colours) are known as '''Ramsey numbers'''. An upper bound for ''R''(''r'',''s'') can be extracted from the proof of the theorem, and other arguments give lower bounds. (The first lower bound was obtained by [[Paul Erdős]] using the [[probabilistic method]].) However, there is a vast gap between the tightest lower bounds and the tightest upper bounds. Consequently, there are very few numbers ''r'' and ''s'' for which we know the exact value of ''R''(''r'',''s''). Computing a lower bound ''L'' for ''R''(''r'',''s'') usually requires exhibiting a blue/red colouring of the graph ''K''<sub>''L''−1</sub> with no blue ''K<sub>r</sub>'' subgraph and no red ''K''<sub>''s''</sub> subgraph. Upper bounds are often considerably more difficult to establish: one either has to check all possible colourings to confirm the absence of a counterexample, or to present a mathematical argument for its absence. A sophisticated computer program does not need to look at all colourings individually in order to eliminate all of them; nevertheless it is a very difficult computational task that existing software can only manage on small sizes. The complexity for searching all possible graphs (via [[brute-force search|brute force]]) is [[Big O notation|O]](2<sup>(''n''−1)(''n''−2)/2</sup>) for an upper bound of ''n'' nodes.<ref>[http://www.learner.org/channel/courses/mathilluminated/units/2/textbook/06.php 2.6 Ramsey Theory from Mathematics Illuminated]</ref>
+As described above, ''R''(3,3) = 6. It is easy to prove that ''R''(4,2) = 4, and, more generally, that ''R''(''s'',2) = ''s'' for all ''s'': a graph on ''s''&nbsp;−&nbsp;1 nodes with all edges coloured red serves as a counterexample and proves that ''R''(''s'',2)&nbsp;≥&nbsp;''s'' ; among colourings of a graph on ''s'' nodes, the colouring with all edges coloured red contains a ''s''-node red subgraph, and all other colourings contain a 2-node blue subgraph (that is, a pair of nodes connected with a blue edge.) Using induction inequalities, it can be concluded that ''R''(4,3)&nbsp;≤&nbsp;''R''(4,2)&nbsp;+&nbsp;''R''(3,3)&nbsp;−&nbsp;1&nbsp;=&nbsp;9, and therefore ''R''(4,4)&nbsp;≤&nbsp;''R''(4,3)&nbsp;+&nbsp;''R''(3,4)&nbsp;≤&nbsp;18. There are only two (4,4,16) graphs (that is, 2-colourings of a complete graph on 16 nodes without 4-node red or blue complete subgraphs) among 6.4×10<sup>22</sup> different 2-colourings of 16-node graphs, and only one (4,4,17) graph (the [[Paley graph]] of order 17) among 2.46×10<sup>26</sup> colourings.<ref>{{cite web|url=http://cs.anu.edu.au/~bdm/data/ramsey.html|title=Ramsey Graphs}}</ref> (This was proven by Evans, Pulham and Sheehan in 1979.) It follows that ''R''(4,4)&nbsp;=&nbsp;18.
+The fact that ''R''(4,5)=25 was first established by [[Brendan McKay]] and [[Stanisław Radziszowski]] in 1995.<ref>{{cite journal|title=R(4,5)&nbsp;=&nbsp;25|author=Brendan D. McKay, Stanislaw P. Radziszowski|journal=Journal of Graph Theory|date=May 1995}}</ref>
+The exact value of ''R''(5,5) is unknown, although it is known to lie between 43 (Geoffrey Exoo) and 49 (McKay and Radziszowski) (inclusive).
+{{cquote|[[Paul Erdős|Erdős]] asks us to imagine an alien force, vastly more powerful than us, landing on Earth and demanding the value of ''R''(5,5) or they will destroy our planet. In that case, he claims, we should marshal all our computers and all our mathematicians and attempt to find the value. But suppose, instead, that they ask for ''R''(6,6). In that case, he believes, we should attempt to destroy the aliens.|||[[Joel Spencer]]<ref>{{citation|title=Ten Lectures on the Probabilistic Method|page=4|author=[[Joel H. Spencer]]|year=1994|publisher=[[Society for Industrial and Applied Mathematics|SIAM]]|isbn=978-0-89871-325-1}}</ref>}}
+McKay, Radziszowski and Exoo employed computer-assisted graph generation methods to conjecture in 1997 that ''R''(5,5) is exactly 43. They were able to construct exactly 656 (5,5,42) graphs, arriving at the same set of graphs through different routes. None of the 656 graphs can be extended to a (5,5,43) graph.<ref>{{cite journal| url=http://cs.anu.edu.au/~bdm/papers/r55.pdf| title=Subgraph Counting Identities and Ramsey Numbers|author=Brendan D. McKay, Stanisław P. Radziszowski|journal=Journal of Combinatorial Theory}}</ref>
+For ''R''(''r'',''s'') with ''r'', ''s''&nbsp;>&nbsp;5, only weak bounds are available. Lower bounds for ''R''(6,6) and ''R''(8,8) have not been improved since 1965 and 1972, respectively.<ref name="Survey">{{cite web|url=http://www.combinatorics.org/ojs/index.php/eljc/issue/view/Surveys|title=Dynamic Surveys}}</ref>
+''R''(''r'',''s'') for values of ''r'' and ''s'' up to 10 are shown in the table below. Where the exact value is unknown, the table lists the best known bounds. ''R''(''r'',''s'') for values of ''r'' and ''s'' less than 3 are given by ''R''(1,''s'') = 1 and ''R''(2,''s'') =&nbsp;''s'' for all values of ''s''. The standard survey on the development of Ramsey number research has been written by Stanisław Radziszowski.
+{| class="wikitable"
+! ''r'',''s''
+! width="80" | 1
+! width="80" | 2
+! width="80" | 3
+! width="80" | 4
+! width="80" | 5
+! width="80" | 6
+! width="80" | 7
+! width="80" | 8
+! width="80" | 9
+! width="80" | 10
+|-
+! 1
+| '''1'''
+|
+|
+|
+|
+|
+|
+|
+|
+|
+|-
+! 2
+| 1
+| '''2'''
+|
+|
+|
+|
+|
+|
+|
+|
+|-
+! 3
+| 1
+| 3
+| '''6'''
+|
+|
+|
+|
+|
+|
+|
+|-
+! 4
+| 1
+| 4
+| 9
+| '''18'''
+|
+|
+|
+|
+|
+|
+|-
+! 5
+| 1
+| 5
+| 14
+| 25
+| '''43–49'''
+|
+|
+|
+|
+|
+|-
+! 6
+| 1
+| 6
+| 18
+| 36–41
+| 58–87
+| '''102–165'''
+|
+|
+|
+|
+|-
+! 7
+| 1
+| 7
+| 23
+| 49–61
+| 80–143
+| 113–298
+| '''205–540'''
+|
+|
+|
+|-
+! 8
+| 1
+| 8
+| 28
+| 58–84
+| 101–216
+| 132–495
+| 217–1031
+| '''282–1870'''
+|
+|
+|-
+! 9
+| 1
+| 9
+| 36
+| 73–115
+| 126–316
+| 169–780
+| 241–1713
+| 317–3583
+| '''565–6588'''
+|
+|-
+! 10
+| 1
+| 10
+| 40–42
+| 92–149
+| 144–442
+| 179–1171
+| 289–2826
+| 331–6090
+| 581–12677
+| '''798–23556'''
+|}
+Since ''R(r, s) = R(s, r)'', there is a trivial symmetry across the diagonal.
+This table is extracted from a larger table compiled by Stanisław Radziszowski.<ref name="Survey" />
+==Asymptotics==
+The inequality ''R''(''r'', ''s'') ≤ ''R''(''r''&nbsp;−&nbsp;1, ''s'') + ''R''(''r'', ''s''&nbsp;−&nbsp;1) may be applied inductively to prove that
+:<math>R(r,s) \leq \binom{r+s-2}{r-1}. </math>
+In particular, this result, due to [[Paul Erdős|Erdős]] and [[George Szekeres|Szekeres]], implies that when ''r'' = ''s'',
+:<math>R(s,s) \leq (1 + o(1))\frac{4^{s-1}}{\sqrt{\pi s}}.</math>
+An exponential lower bound,
+:<math>R(s,s) \geq (1 + o(1)) \frac{s}{\sqrt{2} e} 2^{\frac{s}{2}},</math>
+was given by Erdős in 1947 and was instrumental in his introduction of the probabilistic method. There is obviously a huge gap between these two bounds: for example, for ''s''&nbsp;=&nbsp;10, this gives 101 ≤ ''R''(10,10) ≤ 48620. Nevertheless, exponential growth factors of either bound have not been improved to date and still stand at 4 and <math>\sqrt{2}</math> respectively. There is no known explicit construction producing an exponential lower bound. The best known lower and upper bounds for diagonal Ramsey numbers currently stand at
+:<math>(1 + o(1)) \frac{\sqrt{2} s}{e} 2^{\frac{s}{2}} \leq R(s,s) \leq s^{-\frac{c \log s}{\log \log s}} 4^{s},</math>
+due to [[Joel Spencer|Spencer]] and Conlon respectively.
+For the off-diagonal Ramsey numbers ''R''(3,''t''), it is known that they are of order <math>\tfrac{t^2}{\log t}</math>; this may be stated equivalently as saying that the smallest possible [[Independent set (graph theory)|independence number]] in an ''n''-vertex [[triangle-free graph]] is <math>\Theta(\sqrt{n\log n})</math>. The upper bound for ''R''(3,''t'') is given by [[Miklós Ajtai|Ajtai]], [[János Komlós (mathematician)|Komlós]], and [[Endre Szemerédi|Szemerédi]], the lower bound by [[Jeong Han Kim|Kim]]. More generally, for off-diagonal Ramsey numbers ''R''(''s'', ''t'') with ''s'' fixed and ''t'' growing, the best known bounds are
+:<math> c'_s \frac{t^{\frac{s+1}{2}}}{(\log t)^{\frac{s+1}{2} - \frac{1}{s-2}}} \leq R(s,t) \leq c_s \frac{t^{s-1}}{(\log t)^{s-2}},</math>
+due to Bohman and Keevash and [[Miklós Ajtai|Ajtai]], [[János Komlós (mathematician)|Komlós]] and [[Endre Szemerédi|Szemerédi]] respectively.
+==A multicolour example: ''R''(3,3,3) = 17==
+[[Image:Sixteens.gif|right|thumb|The only two 3-colourings of K<sub>16</sub> with no monochromatic K<sub>3</sub>. The untwisted colouring (top) and the twisted colouring (bottom).]]
+A multicolour Ramsey number is a Ramsey number using 3 or more colours. There is only one nontrivial multicolour Ramsey number for which the exact value is known, namely ''R''(3,3,3) = 17.
+Suppose that you have an edge colouring of a complete graph using 3 colours, red, yellow and green. Suppose further that the edge colouring has no monochromatic triangles. Select a vertex ''v''. Consider the set of vertices that have a green edge to the vertex ''v''. This is called the green neighborhood of ''v''. The green neighborhood of ''v'' cannot contain any green edges, since otherwise there would be a green triangle consisting of the two endpoints of that green edge and the vertex ''v''. Thus, the induced edge colouring on the green neighborhood of ''v'' has edges coloured with only two colours, namely yellow and red. Since ''R''(3,3) = 6, the green neighborhood of ''v'' can contain at most 5 vertices. Similarly, the red and yellow neighborhoods of ''v'' can contain at most 5 vertices each. Since every vertex, except for ''v'' itself, is in one of the green, red or yellow neighborhoods of ''v'', the entire complete graph can have at most 1 + 5 + 5 + 5 = 16 vertices. Thus, we have ''R''(3,3,3) ≤ 17.
+To see that ''R''(3,3,3) ≥ 17, it suffices to draw an edge colouring on the complete graph on 16 vertices with 3 colours that avoids monochromatic triangles. It turns out that there are exactly two such colourings on ''K''<sub>16</sub>, the so-called untwisted and twisted colourings. Both colourings are shown in the figure to the right, with the untwisted colouring on the top, and the twisted colouring on the bottom. In both colourings in the figure, note that the vertices are labeled, and that the vertices ''v''<sub>11</sub> through ''v''<sub>15</sub> are drawn twice, on both the left and the right, in order to simplify the drawings.
+Thus, ''R''(3,3,3) = 17.
+[[Image:Clebsch graph.svg|thumb|right|[[Clebsch graph]]]]If you select any colour of either the untwisted or twisted colouring on ''K''<sub>16</sub>, and consider the graph whose edges are precisely those edges that have the specified colour, you will get the [[Clebsch graph]].
+It is known that there are exactly two edge colourings with 3 colours on ''K''<sub>15</sub> that avoid monochromatic triangles, which can be constructed by deleting any vertex from the untwisted and twisted colourings on ''K''<sub>16</sub>, respectively.
+It is also known that there are exactly 115 edge colourings with 3 colours on ''K''<sub>14</sub> that avoid monochromatic triangles, provided that we consider edge colourings that differ by a permutation of the colours as being the same.
+==Extensions of the theorem==
+The theorem can also be extended to [[hypergraph]]s. An ''m''-hypergraph is a graph whose "edges" are sets of ''m'' vertices – in a normal graph an edge is a set of 2 vertices. The full statement of Ramsey's theorem for hypergraphs is that for any integers ''m'' and ''c'', and any integers ''n''<sub>1</sub>,...,''n''<sub>''c''</sub>, there is an integer ''R''(''n''<sub>1</sub>, ..., ''n''<sub>''c''</sub>;''c'',''m'') such that if the hyperedges of a complete ''m''-hypergraph of order ''R''(''n''<sub>1</sub>,...,''n''<sub>''c''</sub>;''c'',''m'') are coloured with ''c'' different colours, then for some ''i'' between 1 and ''c'', the hypergraph must contain a complete sub-''m''-hypergraph of order ''n''<sub>''i''</sub> whose hyperedges are all colour ''i''. This theorem is usually proved by induction on ''m'', the 'hyper-ness' of the graph. The base case for the proof is ''m''&nbsp;=&nbsp;2, which is exactly the theorem above.
+==Infinite Ramsey theorem==
+A further result, also commonly called ''Ramsey's theorem'', applies to infinite graphs. In a context where finite graphs are also being discussed it is often called the "Infinite Ramsey theorem". As intuition provided by the pictorial representation of a graph is diminished when moving from finite to infinite graphs, theorems in this area are usually phrased in [[set theory|set-theoretic]] terminology.
+<blockquote>'''Theorem.''' Let ''X'' be some [[countable set|countably]] infinite set and colour the elements of ''X''<sup>(''n'')</sup> (the subsets of ''X'' of size ''n'') in ''c'' different colours. Then there exists some infinite subset ''M'' of ''X'' such that the size ''n'' subsets of ''M'' all have the same colour.</blockquote>
+'''Proof''': The proof is given for ''c''&nbsp;=&nbsp;2. It is easy to prove the theorem for an arbitrary number of colours using a 'colour-blindness' argument as above. The proof is by (complete) induction on ''n'', the size of the subsets. For ''n''&nbsp;=&nbsp;1,the statement is equivalent to saying that if you split an infinite set into two sets, one of them is infinite. This is evident. Assuming the theorem is true for ''n'' ≤ ''r'', we prove it for ''n'' = ''r''&nbsp;+&nbsp;1. Given a 2-colouring of the (''r''&nbsp;+&nbsp;1)-element subsets of ''X'', let ''a''<sub>0</sub> be an element of ''X'' and let ''Y'' = ''X''\{''a''<sub>0</sub>}. We then induce a 2-colouring of the ''r''-element subsets of ''Y'', by just adding ''a''<sub>0</sub> to each ''r''-element subset (to get an (''r''&nbsp;+&nbsp;1)-element subset of ''X''). By the induction hypothesis, there exists an infinite subset ''Y''<sub>1</sub> within ''Y'' such that every ''r''-element subset of ''Y''<sub>1</sub> is coloured the same colour in the induced colouring. Thus there is an element ''a''<sub>0</sub> and an infinite subset ''Y''<sub>1</sub> such that all the (''r''&nbsp;+&nbsp;1)-element subsets of ''X'' consisting of ''a''<sub>0</sub> and ''r'' elements of ''Y''<sub>1</sub> have the same colour. By the same argument, there is an element ''a''<sub>1</sub> in ''Y''<sub>1</sub> and an infinite subset ''Y''<sub>2</sub> of ''Y''<sub>1</sub> with the same properties. Inductively, we obtain a sequence {''a''<sub>0</sub>,''a''<sub>1</sub>,''a''<sub>2</sub>,...} such that the colour of each (''r''&nbsp;+&nbsp;1)-element subset (''a''<sub>''i''(1)</sub>,''a''<sub>''i''(2)</sub>,...,''a''<sub>''i''(''r''&nbsp;+&nbsp;1)</sub>) with ''i''(1) < ''i''(2) < ... < ''i''(''r''&nbsp;+&nbsp;1) depends only on the value of ''i''(1). Further, there are infinitely many values of ''i''(''n'') such that this colour will be the same. Take these ''a''<sub>''i''(''n'')</sub>'s to get the desired monochromatic set.
+==Infinite version implies the finite==
+It is possible to deduce the finite Ramsey theorem from the infinite version by a [[proof by contradiction]]. Suppose the finite Ramsey theorem is false. Then there exist integers ''c'', ''n'', ''T'' such that for every integer ''k'', there exists a ''c''-colouring of <math>[k]^{(n)}</math> without a monochromatic set of size ''T''. Let ''C<sub>k</sub>'' denote the ''c''-colourings of <math>[k]^{(n)}</math> without a monochromatic set of size ''T''.
+For any ''k'', the restriction of a colouring in ''C<sub>k+1</sub>'' to <math>[k]^{(n)}</math> (by ignoring the colour of all sets containing ''k''+1) is a colouring in ''C<sub>k</sub>''. Define <math>C^{1}_k</math> to be the colourings in ''C<sub>k</sub>'' which are restrictions of colourings in ''C<sub>k+1</sub>''. Since ''C<sub>k+1</sub>'' is not empty, neither is <math>C^{1}_k</math>.
+Similarly, the restriction of any colouring in <math>C^{1}_{k+1}</math> is in <math>C^{1}_k</math>, allowing one to define <math>C^2_k</math> as the set of all such restrictions, a non-empty set. Continuing so, define <math>C^{m}_k</math> for all integers ''m'', ''k''.
+Now, for any integer ''k'', <math>C_k\supseteq C^1_k\supseteq C^2_k\supseteq \dots</math>, and each set is non-empty. Furthermore, ''C<sub>k</sub>'' is finite as <math>|C_k|\le c^{\frac{k!}{n!(k-n)!}}</math>. It follows that the intersection of all of these sets is non-empty, and let <math>D_k=C_k\cap C^1_k\cap C^2_k\cap \dots</math>. Then every colouring in ''D<sub>k</sub>'' is the restriction of a colouring in ''D<sub>k+1</sub>''. Therefore, by unrestricting a colouring in ''D<sub>k</sub>'' to a colouring in ''D<sub>k+1</sub>'', and continuing doing so, one constructs a colouring of <math>\mathbb N^{(n)}</math> without any monochromatic set of size ''T''. This contradicts the infinite Ramsey theorem.
+If a suitable topological viewpoint is taken, this argument becomes a standard [[compactness theorem|compactness argument]] showing that the infinite version of the theorem implies the finite version.
+==Directed graph Ramsey numbers==
+It is also possible to define Ramsey numbers for ''directed'' graphs.  (These were introduced by [[Paul Erdős|P. Erdős]] & L. Moser.)  Let ''R''(''n'') be the smallest number ''Q'' such that any complete graph with singly directed arcs (also called a "tournament") and with ≥&nbsp;''Q'' nodes contains an acyclic (also called "transitive") ''n''-node subtournament.
+This is the directed-graph analogue of what (above) has been called ''R''(''n'',''n'';2), the smallest number ''Z'' such that any 2-colouring of the edges of a complete ''un''directed graph with ≥&nbsp;''Z'' nodes, contains a monochromatic complete graph on n nodes.  (The directed analogue of the two possible arc ''colours'' is the two ''directions'' of the arcs, the analogue of "monochromatic" is "all arc-arrows point the same way," i.e. "acyclic.")
+Indeed many {{Who|date=January 2010}} find the directed graph problem to actually be ''more'' elegant than the undirected one. We have ''R''(0)=0, ''R''(1)=1, ''R''(2)=2, ''R''(3)=4, ''R''(4)=8, ''R''(5)=14, ''R''(6)=28, 32≤''R''(7)≤55, and ''R''(8) is again a problem you do not want powerful aliens to pose.
+==See also==
+*[[Paris–Harrington theorem]]
+*[[Sim (pencil game)]]
+*[[Infinite Ramsey theory]]
+*[[Van der Waerden number]]
+==Notes==
+{{reflist}}
+==References==
+*{{citation
+ | last1 = Ajtai | first1 = Miklós | author1-link = Miklós Ajtai
+ | last2 = Komlós | first2 = János | author2-link = János Komlós (mathematician)
+ | last3 = Szemerédi | first3 = Endre | author3-link = Endre Szemerédi
+ | journal = J. Combin. Theory Ser. A
+ | pages = 354–360
+ | title = A note on Ramsey numbers
+ | volume = 29
+ | year = 1980
+ | doi = 10.1016/0097-3165(80)90030-8
+ | issue = 3}}.
+*{{citation
+ | last1 = Bohman | first1 = Tom
+ | last2 = Keevash | first2 = Peter
+ | journal = Invent. Math.
+ | pages = 291–336
+ | title = The early evolution of the H-free process
+ | volume = 181
+ | year = 2010
+ | doi = 10.1007/s00222-010-0247-x
+ | issue = 2}}.
+*{{citation
+ | last = Conlon | first = D.
+ | journal = Annals of Mathematics
+ | pages = 941–960
+ | title = A new upper bound for diagonal Ramsey numbers
+ | volume = 170
+ | year = 2009
+ | doi = 10.4007/annals.2009.170.941
+ | issue = 2}}.
+* {{cite journal| first1=Norman | last1=Do | title= Party problems and Ramsey theory
+| journal= Austr. Math. Soc. Gazette | year=2006 | volume=33 | number=5 | pages=306-312
+| url=http://www.austms.org.au/Publ/Gazette/2006/Nov06/Nov06.pdf#page=48
+}}
+*{{citation
+ | last = Erdős | first = Paul | author-link = Paul Erdős
+ | journal = Bull. Amer. Math. Soc.
+ | pages = 292–294
+ | title = Some remarks on the theory of graphs
+ | volume = 53
+ | year = 1947
+ | doi = 10.1090/S0002-9904-1947-08785-1
+ | issue = 4}}.
+*{{citation
+ | last1 = Erdős | first1 = Paul | author1-link = Paul Erdős
+ | last2 = Szekeres | first2 = George | author2-link = George Szekeres
+ | journal = [[Compositio Mathematica]]
+ | pages = 463–470
+ | title = A combinatorial problem in geometry
+ | volume = 2
+ | year = 1935
+ | doi = 10.1007/978-0-8176-4842-8_3
+ | isbn = 978-0-8176-4841-1}}.
+*{{citation
+ | last = Exoo | first = G.
+ | journal = Journal of Graph Theory
+ | pages = 97–98
+ | title = A lower bound for R(5,5)
+ | volume = 13
+ | year = 1989
+ | doi = 10.1002/jgt.3190130113}}.
+*{{citation
+ | last1 = Graham | first1 = R. | author1-link = Ronald Graham
+ | last2 = Rothschild | first2 = B.
+ | last3 = Spencer | first3 = J. H. | author3-link = Joel Spencer
+ | location = New York
+ | publisher = John Wiley and Sons
+ | title = Ramsey Theory
+ | year = 1990}}.
+*{{citation
+ | doi =10.1112/plms/s2-30.1.264
+ | last = Ramsey | first = F. P. | author-link = Frank P. Ramsey
+ | journal = Proceedings of the London Mathematical Society
+ | pages = 264–286
+ | title = On a problem of formal logic
+ | volume = 30
+ | year = 1930}}.
+*{{citation
+ | last = Spencer | first = J. |author-link = Joel H. Spencer
+ | journal = J. Combin. Theory Ser. A
+ | pages = 108–115
+ | title = Ramsey's theorem – a new lower bound
+ | volume = 18
+ | year = 1975
+ | doi = 10.1016/0097-3165(75)90071-0}}.
+*{{citation
+ | last1 = Bian | first1 = Zhengbing
+ | last2 = Chudak | first2 = Fabian
+ | last3 = Macready | first3 = William G.
+ | last4 = Clark | first4 = Lane
+ | last5 = Gaitan | first5 = Frank
+ | title = Experimental determination of Ramsey numbers
+ | journal = Physical Review Letters
+ | volume = 111
+ | pages = 130505-1 - 130505-6
+ | year = 2013
+ | doi = 10.1103/PhysRevLett.111.130505}}.
+==External links==
+{{Wikibooks|Combinatorics|Bounds for Ramsey numbers|Ramsey numbers}}
+* {{springer|title=Ramsey theorem|id=p/r077240}}
+* [http://www.ramseyathome.com/ramsey/ Ramsey@Home] is a [[distributed computing]] project designed to find new lower bounds for various Ramsey numbers using a host of different techniques.
+* [http://www.combinatorics.org/Surveys/ds1/sur.pdf Radziszowski's survey of small Ramsey numbers]
+* this [http://www.cs.bham.ac.uk/~mmk/demos/ramsey-upper-limit.lisp lisp code] computes upper bounds for particular Ramsey numbers given the general rules in Radziszowski
+* [http://RangeVoting.org/PuzzRamsey.html Survey of directed-graph Ramsey numbers]
+* [http://mathworld.wolfram.com/RamseyNumber.html Ramsey Number – from MathWorld] (contains lower and upper bounds up to R(19,19))
+* [http://ginger.indstate.edu/ge/RAMSEY/index.html Ramsey Number – Geoffrey Exoo] (Contains R(5,5) > 42 counter-proof)
+* [http://www.math.sinica.edu.tw/post-doctor/cariolaro/r36.pdf A proof that R(3,6) = 18]
+[[Category:Ramsey theory]]
+[[Category:Theorems in graph theory]]
+[[Category:Articles containing proofs]]

Chebyshev polynomials: Difference between revisions

Revision as of 11:46, 29 January 2014

Example: R(3,3) = 6

Proof of the theorem

2-colour case

General case

Ramsey numbers

Asymptotics

A multicolour example: R(3,3,3) = 17

Extensions of the theorem

Infinite Ramsey theorem

Infinite version implies the finite

Directed graph Ramsey numbers

See also

Notes

References

External links

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