A Subexponential Upper Bound for van der Waerden Numbers $W(3,k)$
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 $$N\le \exp(O(k^{1-c}))\,.$$
2013 ◽
Vol 09
(04)
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pp. 813-843
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2018 ◽
Vol 155
(1)
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pp. 126-163
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2016 ◽
Vol 160
(3)
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pp. 477-494
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