A short proof that $w(3,k) \ge (1-o(1))k^2$

TL;DR

This paper provides a concise proof demonstrating that the two-color van der Waerden number w(3,k) grows at least as fast as approximately k squared, highlighting a significant lower bound in combinatorics.

Contribution

The paper introduces a short proof establishing a near-quadratic lower bound for w(3,k), offering new techniques despite prior known superpolynomial bounds.

Findings

w(3,k) e2 (1-o(1))k^2

Provides a concise proof of the lower bound

Highlights novel techniques in combinatorial proofs

Abstract

Here we present a short proof that the two-color van der Waerden number $w(3,k)$ is bounded from below by $(1-o(1))k^2$. Previous work has already shown that a superpolynomial lower bound holds for $w(3,k)$. However, we believe our result is still is of interest due to our techniques.