New lower bounds for van der Waerden numbers

TL;DR

This paper establishes new lower bounds for van der Waerden numbers by constructing specific colorings that avoid certain arithmetic progressions, significantly improving known bounds and challenging previous conjectures.

Contribution

It introduces novel lower bounds for van der Waerden numbers using advanced combinatorial constructions, surpassing prior estimates.

Findings

Constructed colorings with no 3-term blue progressions

Established lower bounds for w(3,k) of order k^{b(k)}

Challenged previous conjecture that w(3,k) = O(k^2)

Abstract

We show that there is a red-blue colouring of $[N]$ with no blue 3-term arithmetic progression and no red arithmetic progression of length $e^{C(\log N)^{3/4}(\log \log N)^{1/4}}$. Consequently, the two-colour van der Waerden number $w(3,k)$ is bounded below by $k^{b(k)}$, where $b(k) = c \big( \frac{\log k}{\log\log k} \big)^{1/3}$. Previously it had been speculated, supported by data, that $w(3,k) = O(k^2)$.