|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| {{See introduction|Ramsey theory}}
| | Would I be right in assuming you are fed up with all the pain, swelling and itching? And which you would be forever happy if the hemorrhoid bleeding was a thing of the past? You couldn't care less if the hemorrhoids were called outside hemorrhoids, internal hemorrhoids, or anything between can you?<br><br>If you have decided that creams are the [http://hemorrhoidtreatmentfix.com/internal-hemorrhoids internal hemorrhoids] you'd like then you'd find which they are found in numerous drug shops plus pharmacies, plus which are reasonably inexpensive.<br><br>There are actually many treatments which could be employed for hemorrhoid. The first and the most well known is the cream and ointment. These are to be rubbed onto the affected part of the anus. It assists to soothe the already inflamed blood vessels and a momentary relief is accomplished. There is a relaxation of the cells of the rectal column thus far the hemorrhoid is not so much bulged. If there is a bulge still, the pain relief could not do so much to help.<br><br>Hemorrhoid Surgery - This really is only recommended if your hemorrhoids are so serious that no over the counter or house remedy is providing you any type of relief.<br><br>Witch Hazel whenever utilized because an component inside hemorrhoid treatments has an almost magical impact. If you empty a tiny bottle of Witch Hazel into a toilet tissue box, the tissue can absorb it. This makes a pretty soothing substitute for coarser bathroom tissue plus is terrific for gentle cleansing of the delicate region plus reducing pain and itching.<br><br>First off, let's give a superior description of what a sitz bath tub is. A sitz shower tub is a small tub that is especially made to treat the anal region found on the butt of the individual. Like the name states, the individual 'sits' in the tub. The tub contains a little spray nozzle which is in the center of the container, that is connected by a tiny tube, that is connected to either a little bag or is attached to a faucet. The bag would usually contain warm water plus is squeezed by the individual to spray the afflicted area.<br><br>You equally have the choice of following surgery that involves a couple of procedures like using rubber banding or laser, infra red severing of the hemorrhoid. The surgical procedures are constantly significant end alternatives plus is advised only whenever all else has failed. It all depends on what you can tolerate as a person. For me, I usually like to test safer choices before resorting to operation. |
| | |
| 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 × 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 ''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) > 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 [[Mathematical induction|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.
| |
| | |
| <blockquote>'''Lemma 1.''' ''R''(''r'', ''s'') ≤ ''R''(''r'' − 1, ''s'') + ''R''(''r'', ''s'' − 1):</blockquote>
| |
| | |
| '''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<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'' − 1, ''s'') and ''R''(''r'', ''s'' − 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'' − 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.
| |
| | |
| <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'' − 1 and ''c'' are the same colour. Thus the graph is now (''c'' − 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'' − 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'' − 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'' − 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'' − 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 6.4×10<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) = 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) = 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'' > 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'') = ''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'' − 1, ''s'') + ''R''(''r'', ''s'' − 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'' = 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'' = 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'' = 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 ''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'' + 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'' + 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'' + 1)-element subset (''a''<sub>''i''(1)</sub>,''a''<sub>''i''(2)</sub>,...,''a''<sub>''i''(''r'' + 1)</sub>) 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''<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 ≥ ''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 ≥ ''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]]
| |