A subexponential upper bound for van der Waerden numbers W(3,k)

Tomasz Schoen

A subexponential upper bound for van der Waerden numbers W(3,k)

Combinatorics 2020-06-05 v1 Number Theory

Authors: Tomasz Schoen

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 $N\le \exp(O(k^{1-c}))\,.$

Keywords

arithmetic progressions bound upper bounds

Cite

@article{arxiv.2006.02877,
  title  = {A subexponential upper bound for van der Waerden numbers W(3,k)},
  author = {Tomasz Schoen},
  journal= {arXiv preprint arXiv:2006.02877},
  year   = {2020}
}

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