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

TL;DR

This paper establishes a significantly improved upper bound for the van der Waerden number W(3,k), demonstrating that large partitions avoiding certain arithmetic progressions are exponentially limited in size.

Contribution

It provides a new subexponential upper bound for W(3,k), advancing understanding of arithmetic progression avoidance in combinatorial number theory.

Findings

Proves N ≤ exp(O(k^{1-c})) for some constant c>0

Improves previous exponential bounds for van der Waerden numbers

Shows limitations on partitions avoiding 3-term and k-term arithmetic progressions

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