Talk:Van der Waerden's theorem

message: A very simple way to prove the Van der Wareden's theorem is to derive it as a corollary of Gallai's Theorem which in turn is a corollary of the Hales–Jewett theorem. The proof runs like this:
:Apply Gallai's Theorem for V = {0,...k}. Now {b, b+c,...b+kc} is monochromatic. Q.E.D.
Note->Here we were looking for an AP of length k. c and b are the positive integers existing by Gallai's so that cV + b is monochromatic.
This should be included in the article. Any comments?--Shahab 10:16, 17 May 2007 (UTC)
replies: text: While that does seem to work (although I'm not entirely familiar with Gallai's Theorem), I would say that it doesn't make sense to prove things "backwards" - although these generalized versions of each can be proven subsequently and independently, I think it makes more sense to prove them from the ground-up, as opposed to using the most general theorem at any given time to plainly and uninsightfully assert the more specific cases. Cheeser1 15:20, 17 May 2007 (UTC)
replies: text: I am only talking about adding the fact that Van der Waerden's theorem is a corollary of Gallai's theorem (or of the Hales-Jewett Theorem) because it is a point worth realizing that all these theorems are a part of general geometric picture. You are right Cheeser1 when you say that it makes sense to give an insightful picture but there is no reason why the Hales Jewett bit can't be included in the article. Adding some info won't hurt the insight. BTW although the Hales Jewett Theorem is a generalization, it and the Van der Waerden's theorem have very similar colour focussing proofs, so HJ is really not that far advanced then VDW. Cheers--Shahab 13:01, 18 May 2007 (UTC)
replies: text: I'm not sure when it was added, but I just noticed there is mention of both Szemeredi's theorem and Hales-Jewett as stronger theorems. I also noticed that this article's a bit of a mess, I think. Maybe I'll tidy it up.  --Cheeser1 22:21, 17 June 2007 (UTC)
thread_title: Proof

message: Is there anyone who can look at the expression
: V(r,k) \leq 2^{2^{r^{2^{2^{k + 9}}}}},
without instantly see why the comma really needs to be INSIDE the "displayed" TeX?  On the browser I'm using, it is positioned ridiculously. Michael Hardy 01:43, 21 June 2007 (UTC)
thread_title:  importan nitpicking 

message: What is the point in the sentence "The best-known lower bound for V(2,k) is that V(2,k) > 2k / kε for all positive ε.[4]"? It seems to mean that V(2,k) is at least equal to 2k.  —Preceding unsigned comment added by 193.50.42.4 (talk) 09:31, 22 May 2008 (UTC)
replies: text: The statement is that  V(2, k) > 2^k/k^\epsilon for all positive ε.  This means that although V(2, k) is not known to exceed 2^k in general, it is known to come very, very close: it exceeds 2^k / k^{0.000001} and other quantities that are just a hair smaller than 2^k. -- Dominus (talk) 17:10, 22 May 2008 (UTC)

text: In the reference, the statement is exactly " given ε>0 V(2, k) > 2^k/k^\epsilon for all sufficiently large k" which I personally find more illuminating. I was disappointed to find that the reference does not provide a proof of this statement though. If someone knows a source which provides a proof, I think that would be a better thing to reference.76.193.189.206 (talk) 17:33, 8 May 2011 (UTC)
thread_title:  request explanation of bound 

message: The proof that W(2, 3) ≤ 325 seems unnecessary complicated to me.
Consider the following proof that W(2, 3) ≤ 13:
In 5 consecutive two-colored integers two have the same color and have an odd number of integers between them (call them a and b a—Preceding unsigned comment added by 160.45.6.96 (talk) 12:58, 21 April 2010 (UTC)
thread_title:  proof overly complicated 

message: The proof is a bit awkward to read. If MinN is to be a function, it should be coded as \operatorname{MinN} or \mathop{\rm MinN} or at least {\rm MinN} [I removed the < math > tags on purpose in a subsequent edit]. I don't understand why a central star * appears in various places. (I did not read the text in detail but don't think any convolution is involved, and it probably just means multiplication which is traditionally denoted by \times or \cdot.)  Also the line breaking and the u=1,...,L(?) in the line after the equation, rather than directly at the end of (but inside) the equation, is confusing. All in all, it gives the impression that a person not very familiar with mathematical writing was at work here, which make the whole section lose a lot its credibility. — MFH:Talk 20:53, 31 March 2013 (UTC)
thread_title:  typesetting in the proof 

message: The 2nd sentence reads at the moment as follows: 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 whose elements are of the same color.  
The opening of the article would a little more clear and make the connection to Ramsey Theory more clear if it read as follows: 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 in an arbitrary manner using r different colors, then there exist at least k integers of {1, 2, ..., N} in an arithmetic progression whose elements are of the same color. Odahviing (talk) 08:07, 31 August 2019 (UTC)
thread_title:  Opening sentences describing the statement of vdW 

message: The following table summarizes the known values or lower estimates of the first few van der Waerden numbers.
k/r	2 colors	3 colors	4 colors
3  	9       	27      	76
4  	35      	293     	> 1048
5  	178     	> 2173  	> 17705
6  	1132    	> 11191 	> 91331
Indeed, the explicit value of the van der Waerden number W(r,k) is not known for almost all ordered pairs (r,k). The standard combinatorial proof of van der Waerden's theorem only employs the establishment of upper bounds to van der Waerden numbers, and in most cases, these upper bounds are exceedingly weak. For the case r = 2 and k = 3 considered above, a combinatorial argument given below shows that it is sufficient to color the integers {1, ..., 325} with two colors to guarantee there will be a monochromatic arithmetic progression of length three.
For r = 3 and k = 3, the bound given by the standard combinatorial approach is is 7(2·37 + 1)(2·37·(2·37 + 1) + 1), or approximately 4.22·1014616. Employing exhuastive search on 327 = 7.62·1012 candidates, in principle, one could show that any coloring the first twenty seven integers with the colors red, blue and yellow will inevitably lead to at least one monochromatic arithmetic progression of length three. Additionally, it is possible to color {1, ..., 26} with three colors so that there is no monochromatic arithmetic progression of length three arises; for example:
 1 	 2 	 3 	 4 	 5 	 6 	 7 	 8 	 9 	 10 	 11 	 12 	 13 	 14 	 15 	 16 	 17 	 18 	 19 	 20 	 21 	 22 	 23 	 24 	 25 	 26 
 R 	 R 	 Y 	 Y 	 R 	 R 	 Y 	 B 	 Y 	 B  	 B  	 R  	 B  	 R  	 R  	 Y  	 R  	 Y  	 Y  	 B  	 R  	 B  	 B  	 Y  	 B  	 Y 
This proves that W(3,3) =27. 
The best upper bound currently known is due to Timothy Gowers, 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 American mathematician Ronald Graham has offered a prize of US$1000 for a proof of the specific upper bound
: W(2,k) \leq 2^{k^2}.
The best lower bound currently known for W(2, k) is that for all positive \varepsilon we have W(2, k) > 2^k/k^\varepsilon, for all sufficiently large k.Odahviing (talk) 10:02, 31 August 2019 (UTC)
thread_title: Open Problem