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Van der Waerden's theorem is a theorem in the branch of mathematics called Ramsey theory. Van der Waerden's theorem states that for any given positive integers r and k, there is some number N such that if the integers {1, 2, ..., N} are colored, each with one of r different colors, then there are at least k integers in arithmetic progression all of the same color. The least such N is the Van der Waerden number W(r, k). It is named after the Dutch mathematician B. L. van der Waerden.^[1]

For example, when r = 2, you have two colors, say red and blue. W(2, 3) is bigger than 8, because you can color the integers from {1, ..., 8} like this:

       1  2  3  4  5  6  7  8
       B  R  R  B  B  R  R  B

and no three integers of the same color form an arithmetic progression. But you can't add a ninth integer to the end without creating such a progression. If you add a red 9, then the red 3, 6, and 9 are in arithmetic progression. Alternatively, if you add a blue 9, then the blue 1, 5, and 9 are in arithmetic progression. In fact, there is no way of coloring 1 through 9 without creating such a progression. Therefore, W(2, 3) is 9.

It is an open problem to determine the values of W(r, k) for most values of r and k. The proof of the theorem provides only an upper bound. For the case of r = 2 and k = 3, for example, the argument given below shows that it is sufficient to color the integers {1, ..., 325} with two colors to guarantee there will be a single-colored arithmetic progression of length 3. But in fact, the bound of 325 is very loose; the minimum required number of integers is only 9. Any coloring of the integers {1, ..., 9} will have three evenly spaced integers of one color.

For r = 3 and k = 3, the bound given by the theorem is 7(2·3⁷ + 1)(2·3^{7·(2·3⁷ + 1)} + 1), or approximately 4.22·10¹⁴⁶¹⁶. But actually, you don't need that many integers to guarantee a single-colored progression of length 3; you only need 27. (And it is possible to color {1, ..., 26} with three colors so that there is no single-colored arithmetic progression of length 3; for example, RRYYRRYBYBBRBRRYRYYBRBBYBY.)

Anyone who can reduce the general upper bound to any 'reasonable' function can win a large cash prize. Ronald Graham has offered a prize of US$1000 for showing W(2,k)<2^k².^[2] The best-known upper bound is due to Timothy Gowers,^[3] who establishes

W(r,k)\leq 2^{2^{r^{2^{2^{k+9}}}}},

by first establishing a similar result for Szemerédi's theorem, which is a stronger version of Van der Waerden's theorem. The previously best-known bound was due to Saharon Shelah and proceeded via first proving a result for the Hales–Jewett theorem, which is another strengthening of Van der Waerden's theorem.

The best-known lower bound for $W(2,k)$ is that $W(2,k)>2^{k}/k^{\epsilon }$ for all positive $\epsilon$ .^[4]

Proof of Van der Waerden's theorem (in a special case)

The following proof is due to Ron Graham and B.L. Rothschild.^[5] Khinchin^[6] gives a fairly simple proof of the theorem without estimating W(r, k).

We will prove the special case mentioned above, that W(2, 3) ≤ 325. Let c(n) be a coloring of the integers {1, ..., 325}. We will find three elements of {1, ..., 325} in arithmetic progression that are the same color.

Divide {1, ..., 325} into the 65 blocks {1, ..., 5}, {6, ..., 10}, ... {321, ..., 325}, thus each block is of the form {b ·5 + 1, ..., b ·5 + 5} for some b in {0, ..., 64}. Since each integer is colored either red or blue, each block is colored in one of 32 different ways. By the pigeonhole principle, there are two blocks among the first 33 blocks that are colored identically. That is, there are two integers b₁ and b₂, both in {0,...,32}, such that

c(b₁·5 + k) = c(b₂·5 + k)

for all k in {1, ..., 5}. Among the three integers b₁·5 + 1, b₁·5 + 2, b₁·5 + 3, there must be at least two that are the same color. (The pigeonhole principle again.) Call these b₁·5 + a₁ and b₁·5 + a₂, where the a_i are in {1,2,3} and a₁ < a₂. Suppose (without loss of generality) that these two integers are both red. (If they are both blue, just exchange 'red' and 'blue' in what follows.)

Let a₃ = 2·a₂ − a₁. If b₁·5 + a₃ is red, then we have found our arithmetic progression: b₁·5 + a_i are all red.

Otherwise, b₁·5 + a₃ is blue. Since a₃ ≤ 5, b₁·5 + a₃ is in the b₁ block, and since the b₂ block is colored identically, b₂·5 + a₃ is also blue.

Now let b₃ = 2·b₂ − b₁. Then b₃ ≤ 64. Consider the integer b₃·5 + a₃, which must be ≤ 325. What color is it?

If it is red, then b₁·5 + a₁, b₂·5 + a₂, and b₃·5 + a₃ form a red arithmetic progression. But if it is blue, then b₁·5 + a₃, b₂·5 + a₃, and b₃·5 + a₃ form a blue arithmetic progression. Either way, we are done.

A similar argument can be advanced to show that W(3, 3) ≤ 7(2·3⁷+1)(2·3^{7·(2·3⁷+1)}+1). One begins by dividing the integers into 2·3^{7·(2·3⁷ + 1)} + 1 groups of 7(2·3⁷ + 1) integers each; of the first 3^{7·(2·3⁷ + 1)} + 1 groups, two must be colored identically.

Divide each of these two groups into 2·3⁷+1 subgroups of 7 integers each; of the first 3⁷ + 1 subgroups in each group, two of the subgroups must be colored identically. Within each of these identical subgroups, two of the first four integers must be the same color, say red; this implies either a red progression or an element of a different color, say blue, in the same subgroup.

Since we have two identically-colored subgroups, there is a third subgroup, still in the same group that contains an element which, if either red or blue, would complete a red or blue progression, by a construction analogous to the one for W(2, 3). Suppose that this element is yellow. Since there is a group that is colored identically, it must contain copies of the red, blue, and yellow elements we have identified; we can now find a pair of red elements, a pair of blue elements, and a pair of yellow elements that 'focus' on the same integer, so that whatever color it is, it must complete a progression.

The proof for W(2, 3) depends essentially on proving that W(32, 2) ≤ 33. We divide the integers {1,...,325} into 65 'blocks', each of which can be colored in 32 different ways, and then show that two blocks of the first 33 must be the same color, and there is a block coloured the opposite way. Similarly, the proof for W(3, 3) depends on proving that

W(3^{7(2\cdot 3^{7}+1)},2)\leq 3^{7(2\cdot 3^{7}+1)}+1.

By a double induction on the number of colors and the length of the progression, the theorem is proved in general.

Proof

A D-dimensional arithmetic progression consists of numbers of the form:

a+i_{1}s_{1}+i_{2}s_{2}...+i_{D}s_{D}

where a is the basepoint, the s's are the different step-sizes, and the i's range from 0 to L-1. A d-dimensional AP is homogenous for some coloring when it is all the same color.

A D-dimensional arithmetic progression with benefits is all numbers of the form above, but where you add on some of the "boundary" of the arithmetic progression, i.e. some of the indices i's can be equal to L. The sides you tack on are ones where the first k i's are equal to L, and the remaining i's are less than L.

The boundaries of a D-dimensional AP with benefits are these additional arithmetic progressions of dimension d-1,d-2,d-3,d-4, down to 0. The 0 dimensional arithmetic progression is the single point at index value (L,L,L,L...,L). A D-dimensional AP with benefits is homogenous when each of the boundaries are individually homogenous, but different boundaries do not have to necessarily have the same color.

Next define the quantity MinN(L, D, N) to be the least integer so that any assignment of N colors to an interval of length MinN or more necessarily contains a homogenous D-dimensional arithmetical progression with benefits.

The goal is to bound the size of MinN. Note that MinN(L,1,N) is an upper bound for Van-Der-Waerden's number. There are two inductions steps, as follows:

1. Assume MinN is known for a given lengths L for all dimensions of arithmetic progressions with benefits up to D. This formula gives a bound on MinN when you increase the dimension to D+1:

let $M={\mathrm {M} inN}(L,D,n)$

{\mathrm {M} inN}(L,D+1,n)\leq M*{\mathrm {M} inN}(L,1,n^{M})

Proof: First, if you have an n-coloring of the interval 1...I, you can define a block coloring of k-size blocks. Just consider each sequence of k colors in each k block to define a unique color. Call this k-blocking an n-coloring. k-blocking an n coloring of length l produces an n^k coloring of length l/k.

So given a n-coloring of an interval I of size M*MinN(L,1,n^M)) you can M-block it into an n^M coloring of length MinN(L,1,n^M). But that means, by the definition of MinN, that you can find a 1-dimensional arithmetic sequence (with benefits) of length L in the block coloring, which is a sequence of blocks equally spaced, which are all the same block-color, i.e. you have a bunch of blocks of length M in the original sequence, which are equally spaced, which have exactly the same sequence of colors inside.

Now, by the definition of M, you can find a d-dimensional arithmetic sequence with benefits in any one of these blocks, and since all of the blocks have the same sequence of colors, the same d-dimensional AP with benefits appears in all of the blocks, just by translating it from block to block. This is the definition of a d+1 dimensional arithmetic progression, so you have a homogenous d+1 dimensional AP. The new stride parameter s_{D+1} is defined to be the distance between the blocks.

But you need benefits. The boundaries you get now are all old boundaries, plus their translations into identically colored blocks, because i_{D+1} is always less than L. The only boundary which is not like this is the 0 dimensional point when $i_{1}=i_{2}=...=i_{D+1}=L$ . This is a single point, and is automatically homogenous.

2. Assume MinN is known for one value of L and all possible dimensions D. Then you can bound MinN for length L+1.

{\mathrm {M} inN}(L+1,D,n)\leq 2{\mathrm {M} inN}(L,n,n)

proof: Given an n-coloring of an interval of size MinN(L,n,n), by definition, you can find an arithmetic sequence with benefits of dimension n of length L. But now, the number of "benefit" boundaries is equal to the number of colors, so one of the homogenous boundaries, say of dimension k, has to have the same color as another one of the homogenous benefit boundaries, say the one of dimension p<k. This allows a length L+1 arithmetic sequence (of dimension 1) to be constructed, by going along a line inside the k-dimensional boundary which ends right on the p-dimensional boundary, and including the terminal point in the p-dimensional boundary. In formulas:

if

a+Ls_{1}+Ls_{2}...+Ls_{D-k}

has the same color as

a+Ls_{1}+Ls_{2}...+Ls_{D-p}

then

a+L*(s_{1}...+s_{D-k})+u*(s_{D-k+1}...+s_{p})

have the same color

u=0,1,2,...,L-1,L

i.e. u makes a sequence of length L+1.

This constructs a sequence of dimension 1, and the "benefits" are automatic, just add on another point of whatever color. To include this boundary point, one has to make the interval longer by the maximum possible value of the stride, which is certainly less than the interval size. So doubling the interval size will definitely work, and this is the reason for the factor of two. This completes the induction on L.

Base case: MinN(1,d,n)=1, i.e. if you want a length 1 homogenous d-dimensional arithmetic sequence, with or without benefits, you have nothing to do. So this forms the base of the induction. The VanDerWaerden theorem itself is the assertion that MinN(L,1,N) is finite, and it follows from the base case and the induction steps.^[5]

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

External links

Proof of Van der Waerden's theorem

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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

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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
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↑ ^5.0 ^5.1 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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[1] 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

[2] 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

[3] 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

[4] 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

[Graham1974-5] 5.0 ^5.1 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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Contents

Proof of Van der Waerden's theorem (in a special case)

Proof

See also

References

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-{{Other uses}}
+'''Van der Waerden's theorem''' is a theorem in the branch of [[mathematics]] called [[Ramsey theory]].  Van der Waerden's theorem states that for any given positive [[integer]]s ''r'' and ''k'', there is some number ''N'' such that if the integers {1, 2, ..., ''N''} are [[Graph coloring|colored]], each with one of ''r'' different colors, then there are at least ''k'' integers in [[arithmetic progression]] all of the same color. The least such ''N'' is the [[Van der Waerden number]] ''W''(''r'',&nbsp;''k'').  It is named after the Dutch mathematician [[Bartel Leendert van der Waerden|B. L. van der Waerden]].<ref>{{cite journal |authorlink=Bartel Leendert van der Waerden |first=B. L. |last=van der Waerden |title={{lang|de|Beweis einer Baudetschen Vermutung}} |journal=Nieuw. Arch. Wisk. |volume=15 |year=1927 |issue= |pages=212–216 }}</ref>
-{{More footnotes|date=April 2010}}
-[[Image:Airplane vortex edit.jpg|thumb|250px|Vortex created by the passage of an aircraft wing, revealed by colored smoke]]
-A '''Vortex''' (''plural:'' vortices) is a [[Rotation|spinning]], often [[Turbulence|turbulent]],
-flow of [[fluid]].  Any [[spiral]] motion with closed [[Streamlines, streaklines and pathlines|streamlines]] is vortex flow. The motion of the fluid swirling rapidly around a [[center (geometry)|center]] is called a vortex. The speed and rate of [[rotation]] of the fluid in a free (irrotational) vortex are greatest at the center, and decrease progressively with distance from the center, whereas the speed of a forced (rotational) vortex is zero at the center and increases proportional to the distance from the center. Both types of vortex exhibit a decrease in pressure towards the center of rotation, though the rate of decrease in pressure in a free vortex is greater than in a forced vortex.
-==Properties==
+For example, when ''r'' = 2, you have two [[:Category:Color|colors]], say [[red|<font color=red>red</font>]] and [[blue|<font color=blue>blue</font>]]. ''W''(2, 3) is bigger than 8, because you can color the integers from {1, ..., 8} like this:
-[[Image:Crow instability contrail.JPG|[[Crow Instability]] [[contrail]] demonstrates vortex|300px|thumb]]
-Vortices display some special properties:
-*The fluid pressure in a vortex is lowest in the center and rises progressively with distance from the center. This is in accordance with [[Bernoulli's Principle]]. The core of a vortex in air is sometimes visible because of a plume of water vapor caused by [[condensation]] in the low pressure and low temperature of the core. The spout of a [[tornado]] is a classic and frightening example of the visible core of a vortex. A [[dust devil]] is also the core of a vortex, made visible by the dust drawn upwards by the turbulent flow of air from ground level into the low pressure core.
+        '''1''' &nbsp;<u>2</u> &nbsp;<u>3</u> &nbsp;'''4''' &nbsp;'''5''' &nbsp;<u>6</u> &nbsp;<u>7</u> &nbsp;'''8'''
-*The core of every vortex can be considered to contain a vortex line, and every particle in the vortex can be considered to be circulating around the vortex line. Vortex lines can start and end at the boundary of the fluid or form closed loops. They cannot start or end in the fluid. (See [[Helmholtz's theorems]].) Vortices readily deflect and attach themselves to a solid surface. For example, a vortex usually forms ahead of the [[Propeller|propeller disk]] or [[jet engine]] of a slow-moving [[Fixed-wing aircraft|airplane]]. One end of the vortex line is attached to the propeller disk or jet engine, but when the airplane is taxiing the other end of the vortex line readily attaches itself to the ground rather than end in midair. The vortex can suck water and small stones into the core and then into the propeller disk or jet engine.
+        '''<font color=blue>B</font>''' &nbsp;<u><font color=red>R</font></u> &nbsp;<u><font color=red>R</font></u> &nbsp;'''<font color=blue>B</font>''' &nbsp;'''<font color=blue>B</font>''' &nbsp;<u><font color=red>R</font></u> &nbsp;<u><font color=red>R</font></u> &nbsp;'''<font color=blue>B</font>'''
-*Two or more vortices that are approximately parallel and circulating in the same direction will merge to form a single vortex. The [[Circulation (fluid dynamics)|circulation]] of the merged vortex will equal the sum of the [[Circulation (fluid dynamics)|circulations]] of the constituent vortices. For example, a sheet of small vortices flows from the trailing edge of the wing or propeller of an airplane when the wing is developing [[Lift (force)|lift]] or the propeller is developing [[thrust]]. In less than one wing [[Chord (aircraft)|chord]] downstream of the trailing edge of the wing these small vortices merge to form a single vortex. If viewed from the tail of the airplane, looking forward in the direction of flight, there is one [[Wingtip vortices|wingtip vortex]] trailing from the left-hand wing and circulating clockwise, and another wingtip vortex trailing from the right-hand wing and circulating anti-clockwise. The result is a region of downwash behind the wing, between the pair of [[wingtip vortices]]. These two [[wingtip vortices]] do not merge because they are circulating in opposite directions.
-*Vortices contain a lot of energy in the circular motion of the fluid.  In an ideal fluid this energy can never be dissipated and the vortex would persist forever. However, real fluids exhibit [[viscosity]] and this dissipates energy very slowly from the core of the vortex. (See [[Rankine vortex]]). It is only through dissipation of a vortex due to viscosity that a vortex line can end in the fluid, rather than at the boundary of the fluid. For example, the [[wingtip vortices]] from an airplane dissipate slowly and linger in the atmosphere long after the airplane has passed. This is a hazard to other aircraft and is known as [[wake turbulence]].
-==Dynamics==
+and no three integers of the same color form an [[arithmetic progression]].  But you can't add a ninth integer to the end without creating such a progression.  If you add a [[red|<font color=red>red 9</font>]], then the [[red|<font color=red>red 3</font>]], [[red|<font color=red>6</font>]], and [[red|<font color=red>9</font>]] are in arithmetic progression.  Alternatively, if you add a [[blue|<font color=blue>blue 9</font>]], then the [[blue|<font color=blue>blue 1</font>]], [[blue|<font color=blue>5</font>]], and [[blue|<font color=blue>9</font>]] are in arithmetic progression.  In fact, there is no way of coloring 1 through 9 without creating such a progression.  Therefore, ''W''(2, 3) is 9.
-A vortex can be any circular or rotary flow. Perhaps unexpectedly, not all vortices possess
-''[[vorticity]]''. Vorticity is a mathematical concept used in [[fluid dynamics]]. It can be related to the amount of "circulation" or "rotation" in a fluid. In fluid dynamics, vorticity is the circulation per unit area at a point in the flow field. It is a [[vector (geometry)|vector]] quantity, whose direction is (roughly speaking) along the axis of the swirl. The vorticity of a free vortex is zero everywhere except at the center, whereas the vorticity of a forced vortex is non-zero. Vorticity is an approximately conserved quantity, meaning that it is not readily created or destroyed in a flow. Therefore, flows that start with minimal vorticity, such as water in a basin, create vortices with minimal vorticity, such as the characteristic swirling and approximately free vortex structure when it drains. By contrast, fluids that initially have vorticity, such as water in a rotating bowl, form vortices with vorticity, exhibited by the much less pronounced low pressure region at the center of this flow. Also in fluid dynamics, the movement of a fluid can be said to be ''[[vortical]]'' if the fluid moves around in a circle, or in a helix, or if it tends to spin around some axis. Such motion can also be called [[solenoidal vector field|solenoidal]]. In the atmospheric sciences, vorticity is a property that characterizes large-scale rotation of air masses. Since the atmospheric circulation is nearly horizontal, the (3 dimensional) vorticity is nearly vertical, and it is common to use the vertical component as a scalar vorticity. Mathematically, vorticity <math>\vec\omega</math> is defined as the [[curl (mathematics)|curl]] of the ''fluid velocity'' <math>\vec{\mathit{u}}</math>:
-: <math> \vec \omega = \nabla \times \vec{\mathit{u}}.</math>
+It is an open problem to determine the values  of ''W''(''r'', ''k'') for most values of ''r'' and ''k''. The proof of the theorem provides only an upper bound.  For the case of ''r'' = 2 and ''k'' = 3, for example, the argument given below shows that it is sufficient to color the integers {1, ..., 325} with two colors to guarantee there will be a single-colored arithmetic progression of length 3. But in fact, the bound of 325 is very loose; the minimum required number of integers is only 9.  Any coloring of the integers {1, ..., 9} will have three evenly spaced integers of one color.
-==Two types of vortex==<!-- This section is linked from [[Vorticity]] -->
+For ''r'' = 3 and ''k'' = 3, the bound given by the theorem is 7(2·3<sup>7</sup>&nbsp;+&nbsp;1)(2·3<sup>7·(2·3<sup>7</sup>&nbsp;+&nbsp;1)</sup>&nbsp;+&nbsp;1), or approximately 4.22·10<sup>14616</sup>. But actually, you don't need that many integers to guarantee a single-colored progression of length 3; you only need 27. (And it is possible to color {1, ..., 26} with three colors so that there is no single-colored arithmetic progression of length 3; for example, RRYYRRYBYBBRBRRYRYYBRBBYBY.)
-In fluid mechanics, a distinction is often made between two limiting vortex cases. One is called the free (irrotational) vortex, and the other is the forced (rotational) vortex. These are considered below, using the following example:
-<center>
-{| class="wikitable" style="width:30%;"
-|+Types of vortex illustrated by the movement of two autumn leaves
-|-
-|[[Image:Vortex south.png|227px]]
-|[[Image:Free vortex east.png|227px]]
-|[[Image:Vortex east.png|217px]]
-|-
-|Reference position in a counter-clockwise vortex.
-|In an ''irrotational'' vortex, the leaves preserve their original orientation while moving counter-clockwise.
-|In a ''rotational'' vortex, the leaves rotate with the counter-clockwise flow.
-|}
-</center>
-===Free (irrotational) vortex===
+Anyone who can reduce the general upper bound to any 'reasonable' function can win a large cash prize. [[Ronald Graham]] has offered a prize of [[US$]]1000 for showing ''W''(2,''k'')&lt;2<sup>''k''<sup>2</sup></sup>.<ref>{{cite journal |authorlink=Ronald Graham |first=Ron |last=Graham |title=Some of My Favorite Problems in Ramsey Theory |journal=INTEGERS (The Electronic Journal of Combinatorial Number Theory |url=http://www.integers-ejcnt.org/vol7-2.html |volume=7 |issue=2 |year=2007 |pages=#A2 }}</ref> The best-known upper bound is due to [[Timothy Gowers]],<ref>{{cite journal |authorlink=Timothy Gowers |first=Timothy |last=Gowers |title=A new proof of Szemerédi's theorem |journal=Geom. Funct. Anal. |volume=11 |issue=3 |pages=465–588 |year=2001 |url=http://www.dpmms.cam.ac.uk/~wtg10/papers.html |doi=10.1007/s00039-001-0332-9 }}</ref> who establishes
-[[File:Irrotational vortex.gif|thumb|right|Irrotational vortex]]When fluid is drawn down a plug-hole, one can observe the phenomenon of a '''free vortex''' or '''line vortex'''. The tangential velocity ''v'' varies inversely as the distance ''r'' from the center of rotation, so the angular momentum ''rv'' is uniform everywhere throughout the flow; the vorticity is zero everywhere (except for a singularity at the center-line) and the [[circulation (fluid dynamics)|circulation]] about a contour containing ''r''&nbsp;=&nbsp;0 has the same value everywhere.<ref name=LJC7.5>Clancy, L.J., ''Aerodynamics'', sub-section 7.5 </ref> The [[free surface]] (if present) dips sharply ([[inverse-square law|as ''r''<sup>&nbsp;−2</sup>]] ) as the center line is approached.
-The tangential velocity is given by:
+: <math>W(r,k) \leq 2^{2^{r^{2^{2^{k + 9}}}}},</math>
-:<math>v_{\theta} = \frac{\Gamma}{2 \pi r}\,</math>
+by first establishing a similar result for [[Szemerédi's theorem]], which is a stronger version of Van der Waerden's theorem.  The previously best-known bound was due to [[Saharon Shelah]] and proceeded via first proving a result for the [[Hales&ndash;Jewett theorem]], which is another strengthening of Van der Waerden's theorem.
-where [[Γ]] is the circulation and r is the radial distance from the center of the vortex.
-In non-technical terms, the fluid near the center of the vortex completes one revolution in a shorter time than the fluid far from the center. The speed of the fluid also decreases as the distance from the center increases. Imagine a leaf floating in a free vortex. The orientation of the leaf remains constant, even though it is moving around the center of the vortex.
+The best-known lower bound for <math>W(2, k)</math> is that <math>W(2, k) > 2^k/k^\epsilon</math> for all positive <math>\epsilon</math>.<ref>{{cite book |title=Discrete Mathematics And Its Applications |editor=M. Sethumadhavan |last=Brown | first=Tom C. | pages=80 | chapter=A partition of the non-negative integers, with applications to Ramsey theory |authorlink= |coauthors= |year=2006 |publisher=Alpha Science Int'l Ltd. |location= |isbn=81-7319-731-8 }}</ref>
-===Forced (rotational) vortex===
+== Proof of Van der Waerden's theorem (in a special case) ==
-[[File:Rotational vortex.gif|thumb|right|Rotational vortex]]
-In a '''forced vortex''' the fluid rotates as a solid body (there is no shear). The motion can be realized by placing a dish of fluid on a turntable rotating at ω radian/s; the fluid has vorticity of 2ω everywhere, and the free surface (if present) is a paraboloid.
-The tangential velocity is given by:<ref name=LJC7.5/>
+The following proof is due to [[Ronald Graham|Ron Graham]] and B.L. Rothschild.<ref name="Graham1974">{{cite journal |authorlink=Ronald Graham |first=R. L. |last=Graham |first2=B. L. |last2=Rothschild |title=A short proof of van der Waerden's theorem on arithmetic progressions |journal=Proc. American Math. Soc. |volume=42 |issue=2 |year=1974 |pages=385–386 |doi=10.1090/S0002-9939-1974-0329917-8 }}</ref> [[A. Ya. Khinchin|Khinchin]]<ref>{{Cite document
+  | last1 = Khinchin   | first1 = A. Ya.
+  | title = Three Pearls of Number Theory
+  | publisher = Dover
+  | location = Mineola, NY
+  | date = 1998
+  | isbn = 978-0-486-40026-6
+  | postscript = .}}
+</ref> gives a fairly simple proof of the theorem without estimating ''W''(''r'',&nbsp;''k'').
-:<math>v_{\theta} = \omega r\,</math>
+We will prove the special case mentioned above, that ''W''(2, 3) ≤ 325. Let ''c''(''n'') be a coloring of the integers {1, ..., 325}.  We will find three elements of {1, ..., 325} in arithmetic progression that are the same color.
-where ω is the [[angular velocity]] and r is the radial distance from the center of the vortex.
-==Vortices in magnets==
+Divide {1, ..., 325} into the 65 blocks {1, ..., 5}, {6, ..., 10}, ... {321, ..., 325}, thus each block is of the form {''b'' ·5 + 1, ..., ''b'' ·5 + 5} for some ''b'' in {0, ..., 64}. Since each integer is colored either red or blue, each block is colored in one of 32 different ways.  By the [[pigeonhole principle]], there are two blocks among the first 33 blocks that are colored identically. That is, there are  two integers ''b''<sub>1</sub> and ''b''<sub>2</sub>, both in {0,...,32}, such that
-Different classes of vortex waves also exist in magnets. There are exact solutions to classical nonlinear magnetic equations e.g. [[Landau–Lifshitz model|Landau-Lifshitz equation]], continuum [[Heisenberg model (classical)|Heisenberg model]], [[Ishimori equation]], [[nonlinear Schrödinger equation]] and so on.
-==Observations==
+: ''c''(''b''<sub>1</sub>&middot;5 + ''k'') = ''c''(''b''<sub>2</sub>&middot;5 + ''k'')
-A vortex can be seen in the spiraling motion of [[air]] or [[liquid]] around a center of [[rotation]]. The circular current of water of conflicting [[tide]]s often form vortex shapes. [[Turbulence|Turbulent flow]] makes many vortices. A good example of a vortex is the [[Earth's atmosphere|atmospheric]] phenomenon of a [[whirlwind]] or a [[tornado]] or [[dust devil]]. This whirling air mass mostly takes the form of a [[helix]], [[column]], or [[spiral]]. Tornadoes develop from severe thunderstorms, usually spawned from [[squall line]]s and [[supercell thunderstorm]]s, though they sometimes happen as a result of a [[hurricane]].
-In atmospheric physics, a ''[[Mesocyclone|mesovortex]]'' is on the scale of a few miles (smaller than a hurricane but larger than a tornado). <sub>[2]</sub> On a much smaller scale, a vortex is usually formed as water goes down a drain, as in a [[sink]] or a [[toilet]]. This occurs in water as the revolving mass forms a [[whirlpool]]. This whirlpool is caused by water flowing out of a small opening in the bottom of a [[sink|basin]] or [[reservoir (water)|reservoir]]. This swirling flow structure within a region of fluid flow opens downward from the water surface.
+for all ''k'' in {1, ..., 5}.  Among the three integers ''b''<sub>1</sub>·5 + 1, ''b''<sub>1</sub>·5 + 2, ''b''<sub>1</sub>·5 + 3, there must be at least two that are the same color. (The [[pigeonhole principle]] again.)  Call these ''b''<sub>1</sub>·5 + ''a''<sub>1</sub> and ''b''<sub>1</sub>·5 + ''a''<sub>2</sub>, where the ''a''<sub>''i''</sub> are in {1,2,3} and ''a''<sub>1</sub> &lt; ''a''<sub>2</sub>.  Suppose (without loss of generality) that these two integers are both red.  (If they are both blue, just exchange 'red' and 'blue' in what follows.)
-===Instances===
+Let ''a''<sub>3</sub> = 2·''a''<sub>2</sub>&nbsp;&minus;&nbsp;''a''<sub>1</sub>. If ''b''<sub>1</sub>·5 + ''a''<sub>3</sub> is red, then we have found our arithmetic progression: ''b''<sub>1</sub>·5&nbsp;+&nbsp;''a''<sub>''i''</sub> are all red.
-*In the [[hydrodynamics|hydrodynamic]] interpretation of the behaviour of [[electromagnetic field]]s, the acceleration of electric fluid in a particular direction creates a positive vortex of magnetic fluid. This in turn creates around itself a corresponding negative vortex of electric fluid.
-*[[Smoke ring]] : A ring of smoke that persists for a surprisingly long time, illustrating the slow rate at which viscosity dissipates the energy of a vortex.
+Otherwise, ''b''<sub>1</sub>·5 + ''a''<sub>3</sub> is blue. Since ''a''<sub>3</sub> ≤ 5,  ''b''<sub>1</sub>·5 + ''a''<sub>3</sub> is in the ''b''<sub>1</sub> block, and since the ''b''<sub>2</sub> block is colored identically, ''b''<sub>2</sub>·5 + ''a''<sub>3</sub> is also blue.
-*[[Bubble ring]] : A ring of bubbles formed under water, moving in any direction, created by some playful dolphins and other whales.
-*[[Lift-induced drag]] of a [[wing]] on an [[aircraft]].
+Now let ''b''<sub>3</sub> = 2·''b''<sub>2</sub>&nbsp;&minus;&nbsp;''b''<sub>1</sub>. Then ''b''<sub>3</sub> ≤ 64. Consider the integer  ''b''<sub>3</sub>·5 + ''a''<sub>3</sub>, which must be ≤ 325. What color is it?
-*The primary cause of [[drag (physics)|drag]] in the [[sail]] of a [[sloop]].
-*[[Whirlpool]]: a swirling body of water produced by ocean tides or by a hole underneath the vortex where the water would drain out, such as a bathtub. A large, powerful whirlpool is known as a [[maelstrom]]. In popular imagination, but only rarely in reality, they can have the dangerous effect of destroying boats. Examples are [[Charybdis]] of classical [[mythology]] in the Straits of [[Messina]], [[Italy]]; the [[Naruto whirlpool]]s of [[Nankaido]], [[Japan]]; the [[Maelstrom]], [[Lofoten]], [[Norway]].
+If it is red, then ''b''<sub>1</sub>·5 + ''a''<sub>1</sub>, ''b''<sub>2</sub>·5 + ''a''<sub>2</sub>, and ''b''<sub>3</sub>·5 + ''a''<sub>3</sub> form a red arithmetic progression. But if it is blue, then ''b''<sub>1</sub>·5 + ''a''<sub>3</sub>, ''b''<sub>2</sub>·5 + ''a''<sub>3</sub>, and ''b''<sub>3</sub>·5 + ''a''<sub>3</sub> form a blue arithmetic progression. Either way, we are done.
-*[[Ice stalactite]]s are formed by a rotating column of downward-moving supercooled brine.
-*A small stream of falling water starts rotating immediately on release and does so until the speed of downward movement overcomes the cohesion of surface tension and causes its breakup into spray.
+A similar argument can be advanced to show that ''W''(3, 3) ≤ 7(2·3<sup>7</sup>+1)(2·3<sup>7·(2·3<sup>7</sup>+1)</sup>+1). One begins by dividing the integers into  2·3<sup>7·(2·3<sup>7</sup>&nbsp;+&nbsp;1)</sup>&nbsp;+&nbsp;1 groups of 7(2·3<sup>7</sup>&nbsp;+&nbsp;1) integers each; of the first 3<sup>7·(2·3<sup>7</sup>&nbsp;+&nbsp;1)</sup>&nbsp;+&nbsp;1 groups, two must be colored identically.
-*[[Tornado]] : a violent windstorm characterized by a twisting, funnel-shaped cloud. A less violent version of a tornado, over water, is called a [[waterspout]].
-*[[Tropical cyclone|Hurricane]] : a much larger, swirling body of clouds produced by evaporating warm ocean water and influenced by the Earth's rotation. Similar, but far greater, vortices are also seen on other planets, such as the permanent [[Great Red Spot]] on [[Jupiter]] and the intermittent [[Great Dark Spot]] on [[Neptune]].
+Divide each of these two groups into 2·3<sup>7</sup>+1 subgroups of 7 integers each; of the first 3<sup>7</sup>&nbsp;+&nbsp;1 subgroups in each group, two of the subgroups must be colored identically.  Within each of these identical subgroups, two of the first four integers must be the same color, say red; this implies either a red progression or an element of a different color, say blue, in the same subgroup.
-*[[Polar vortex]] : a persistent, large-scale cyclone centered near the Earth's poles, in the middle and upper troposphere and the stratosphere.
-*[[Sunspot]] : dark region on the Sun's surface (photosphere) marked by a lower temperature than its surroundings, and intense magnetic activity.
+Since we have two identically-colored subgroups, there is a third subgroup, still in the same group that contains an element which, if either red or blue, would complete a red or blue progression, by a construction analogous to the one for ''W''(2, 3). Suppose that this element is yellow. Since there is a group that is colored identically, it must contain copies of the red, blue, and yellow elements we have identified; we can now find a pair of red elements, a pair of blue elements, and a pair of yellow elements that 'focus' on the same integer, so that whatever color it is, it must complete a progression.
-**[[Alfven waves]]
-*The [[accretion disk]] of a [[black hole]] or other massive gravitational source.
+The proof for ''W''(2, 3) depends essentially on proving that ''W''(32, 2) ≤ 33.  We divide the integers {1,...,325} into 65 'blocks', each of which can be colored in 32 different ways, and then show that two blocks of the first 33 must be the same color, and there is a block coloured the opposite way.  Similarly, the proof for ''W''(3, 3) depends on proving that
-*[[Spiral galaxy]] : a type of galaxy in the [[Hubble sequence]] that is characterized by a thin, rotating disk. Earth's galaxy, the [[Milky Way]], is of this type.
+: <math>W(3^{7(2 \cdot 3^7+1)},2) \leq 3^{7(2 \cdot 3^7+1)}+1.</math>
+By a double [[mathematical induction|induction]] on the number of colors and the length of the progression, the theorem is proved in general.
+== Proof ==
+A [[Generalized arithmetic progression|''D-dimensional arithmetic progression'']] consists of
+numbers of the form:
+::<math> a + i_1 s_1 + i_2 s_2 ... + i_D s_D </math>
+where a is the basepoint, the s's are the different step-sizes, and the i's range from 0 to L-1. A d-dimensional AP is ''homogenous'' for some coloring when it is all the same color.
+A ''D-dimensional arithmetic progression with benefits'' is all numbers of the form above, but where you add on some of the "boundary" of the arithmetic progression, i.e. some of the indices i's can be equal to L. The sides you tack on are ones where the first k i's are equal to L, and the remaining i's are less than L.
+The boundaries of a D-dimensional AP with benefits are these additional arithmetic progressions of dimension d-1,d-2,d-3,d-4, down to 0. The 0 dimensional arithmetic progression is the single point at index value (L,L,L,L...,L). A D-dimensional AP with benefits is ''homogenous'' when each of the boundaries are individually homogenous, but different boundaries do not have to necessarily have the same color.
+Next define the quantity MinN(L, D, N) to be the least integer so
+that any assignment of N colors to an interval of length MinN or more
+necessarily contains a homogenous D-dimensional arithmetical progression with benefits.
+The goal is to bound the size of MinN. Note that MinN(L,1,N) is an upper bound for Van-Der-Waerden's
+number. There are two inductions steps, as follows:
+. Assume MinN is known for a given lengths L for all dimensions of arithmetic progressions with benefits up to D. This formula gives a bound on MinN when you increase the dimension to D+1:
+let <math> M = {\mathrm MinN}(L,D,n)</math>
+::<math> {\mathrm MinN}(L, D+1 , n) \le  M*{\mathrm MinN}(L,1,n^M)</math>
+Proof:
+First, if you have an n-coloring of the interval 1...I, you can define a ''block coloring'' of k-size
+blocks. Just consider each sequence of k colors in each k block to define a unique color. Call this ''k-blocking'' an n-coloring. k-blocking an n coloring of length l produces an n^k coloring of length l/k.
+So given a n-coloring of an interval I of size M*MinN(L,1,n^M)) you can M-block it into an n^M coloring
+of length MinN(L,1,n^M). But that means, by the definition of MinN, that you can find a 1-dimensional arithmetic sequence (with benefits) of length L in the block coloring, which is a sequence of blocks equally spaced, which are all the same block-color, i.e. you have a bunch of blocks of length M in the original sequence, which are equally spaced, which have exactly the same sequence of colors inside.
+Now, by the definition of M, you can find a d-dimensional arithmetic sequence with benefits in any one of these blocks, and since all of the blocks have the same sequence of colors, the same d-dimensional AP with benefits appears in all of the blocks, just by translating it from block to block. This is the definition of a d+1 dimensional arithmetic progression, so you have a homogenous d+1 dimensional AP. The new stride parameter s_{D+1} is defined to be the distance between the blocks.
+But you need benefits. The boundaries you get now are all old boundaries, plus their translations into identically colored blocks, because i_{D+1} is always less than L. The only boundary which is not like this is the 0 dimensional point when <math>i_1=i_2=...=i_{D+1}=L</math>. This is a single point, and is automatically homogenous.
+. Assume MinN is known for one value of L and all possible dimensions D. Then you can bound MinN for length L+1.
+::<math>{\mathrm MinN}(L+1,D,n) \le 2{\mathrm MinN}(L,n,n)</math>
+proof:
+Given an n-coloring of an interval of size MinN(L,n,n), by definition, you can find an arithmetic sequence with benefits of dimension n of length L. But now, the number of "benefit" boundaries is equal to the number of colors, so one of the homogenous boundaries, say of dimension k, has to have the same color as another one of the homogenous benefit boundaries, say the one of dimension p<k. This allows a length L+1 arithmetic sequence (of dimension 1) to be constructed, by going along a line inside the k-dimensional boundary which ends right on the p-dimensional boundary, and including the terminal point in the p-dimensional boundary. In formulas:
+if
+::<math> a+ L s_1 +L s_2... + L s_{D-k}</math> has the same color as
+::<math> a + L s_1 +L s_2 ... +L s_{D-p}</math>
+then
+::<math> a + L*(s_1 ... +s_{D-k}) + u *(s_{D-k+1} ... +s_p) </math> have the same color
+::<math> u = 0,1,2,...,L-1,L </math> i.e. u makes a sequence of length L+1.
+This constructs a sequence of dimension 1, and the "benefits" are automatic, just add on another point of whatever color. To include this boundary point, one has to make the interval longer by the maximum possible value of the stride, which is certainly less than the interval size. So doubling the interval size will definitely work, and this is the reason for the factor of two. This completes the induction on L.
+Base case: MinN(1,d,n)=1, i.e. if you want a length 1 homogenous d-dimensional arithmetic sequence, with or without benefits, you have nothing to do. So this forms the base of the induction. The VanDerWaerden theorem itself is the assertion that MinN(L,1,N) is finite, and it follows from the base case and the induction steps.<ref name="Graham1974" />
 ==See also==
-{{Portal|Physics}}
+* [[Van der Waerden number]]s for all known values for ''W''(''n'',''r'') and the best-known bounds for unknown values
-<div style="column-count:3;-moz-column-count:3;-webkit-column-count:3">
-*[[Artificial gravity]]
-*[[Batchelor vortex]]
-*[[Cyclonic separation]]
-*[[Eddy (fluid dynamics)|Eddy]]
-*[[Gyre]]
-*[[Helmholtz's theorems]]
-*[[History of fluid mechanics]]
-*[[Horseshoe vortex]]
-*[[Kelvin–Helmholtz instability]]
-*[[Quantum vortex]]
-*[[Rankine vortex]]
-*[[Shower-curtain effect]]
-*[[Strouhal number]]
-*[[Viktor Schauberger]]
-*[[Vile Vortices]]
-*[[Von Kármán vortex street]]
-*[[Vortex engine]]
-*[[Vortex ring]]
-*[[Vortex tube]]
-*[[Vortex cooler]]
-*[[Vortex shedding]]
-*[[Vortex stretching]]
-*[[Vortex induced vibration]]
-*[[Vorticity]]
-*[[Whirlpool]]
-*[[Wingtip vortices]]
-*[[Wormhole]]
-</div>
-==Notes==
+==References==
 {{reflist}}
-==References and further reading==
-*"''[http://oap2.weather.com/glossary/v.html Weather Glossary]''"' The Weather Channel Interactive, Inc.. 2004.
-*"''[http://www.bbsr.edu/rpi/meetpart/paper/glossary.html Glossary and Abbreviations]''". Risk Prediction Initiative. The Bermuda Biological Station for Research, Inc.. St. George's, Bermuda. 2004.
-*Loper, David E., "''An analysis of confined magnetohydrodynamic vortex flows''". Case Institute of Technology. Washington, National Aeronautics and Space Administration; for sale by the Clearinghouse for Federal Scientific and Technical Information, Springfield, Va. 1966. (NASA contractor report NASA CR-646) LCCN 67060315
-*[[George Batchelor|Batchelor, G. K.]] (1967), ''An Introduction to Fluid Dynamics'', Cambridge Univ. Press, Ch. 7 et seq
-* {{cite book|last=Falkovich|first=G.|title=Fluid Mechanics, a short course for physicists|url=http://www.cambridge.org/gb/knowledge/isbn/item6173728/?site_locale=en_GB|publisher=Cambridge University Press|year=2011|isbn=978-1-107-00575-4}}
-*Clancy, L.J. (1975), ''Aerodynamics'', Pitman Publishing Limited, London.  ISBN 0-273-01120-0
 ==External links==
-{{Commons category|Vortex}}
+* [http://www.math.uga.edu/~lyall/REU/ramsey.pdf Proof of Van der Waerden's theorem]
-*[http://www.cse.salford.ac.uk/profiles/gsmcdonald/Solitons/Optical_Vortex_Solitons.php Optical Vortices]
-*[http://evgars.com Create Vortex]
-*[http://www.animalu.com/pics/dd1.htm Dust Devil Movie] A short movie showing many spinning vortices of varying sizes
-*[http://www.eng.nus.edu.sg/mpelimtt/collision.mpg Video of two water vortex rings colliding] ([[MPEG]])
-*[http://www.weizmann.ac.il/complex/falkovich/fluid-mechanics Fluid Mechanics website with movies, Q&A, etc]
-*[http://www.bubblerings.com/ BubbleRings.com] Web site on "bubble rings", which are underwater rings made of air formed from vortices. The site has some information on how these rings work.
-*[http://maxwell.ucdavis.edu/~cole/phy9b/notes/fluids_ch3.pdf Chapter 3 Rotational Flows: Circulation and Turbulence]
-*[http://www.youtube.com/watch?v=Yb0r67_I424 Video of Vortex Formation and Annihilation in a Superconductor] (By [http://www.dartmouth.edu/~jthorarinson Joel Thorarinson])
-[[Category:Vortices| ]]
+[[Category:Ramsey theory]]
-[[Category:Aerodynamics]]
+[[Category:Theorems in discrete mathematics]]
-[[Category:Fluid dynamics]]
+[[Category:Articles containing proofs]]
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