Abstract
We show an improved upper estimate for van der Waerden number $W(3,k):$ there is an absolute constant $c>0$ such that if $\{1,\dots,N\}=X\cup Y$ is a partition such that $X$ does not contain any arithmetic progression of length $3$ and $Y$ does not contain any arithmetic progression of length $k$ then
 $$N\le \exp(O(k^{1-c}))\,.$$
Highlights
The van der Waerden number W (k, l) is the smallest positive integer N such that in any partition {1, . . . , N } = X ∪ Y there is an arithmetic progression of length k in X or an arithmetic progression of length l in Y
W (k, l) is related to Szemeredi’s theorem on arithmetic progressions [20] and any effective estimate in this theorem leads to an upper bound on the van der Waerden numbers
W (3, k) N δ−2−c/6 k3 which concludes the proof
Summary
W (k, l) is related to Szemeredi’s theorem on arithmetic progressions [20] and any effective estimate in this theorem leads to an upper bound on the van der Waerden numbers. Green [13] proposed a very clever argument based on arithmetic properties of sumsets to bound W (3, k) Building on this method and applying results from [10] it was shown in [11] that. Our argument is based on the method of [18], which explores in details the structure of a large spectrum This method can be partly applied (see Lemma 5) in our approach and it deals only with a progression-free partition class. That result implies directly that W (3, k) exp(Ck1−c) with c ≈ 2 . −221000
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