An improved lower bound for the van der Waerden number $w(3,k)$

Abstract

For an integer $k\ge 3$, let $w(3,k)$ be the least $n$ such that every red-blue coloring of $[n]$ contains either a nontrivial three-term arithmetic progression in blue or a nontrivial $k$-term arithmetic progression in red. A recent breakthrough of Green proved that $w(3,k)\ge k^{\Omega \left({\left(\frac{\log k}{\log \log k}\right)}^{1/3}\right)}$ when $k$ is large, and Hunter later improved the bound to $w(3,k)\ge k^{\Omega \left(\frac{\log k}{\log \log k}\right)}$. On the other hand, Green remarked that it is reasonable to believe that $w(3,k)\le k^{O(\log k)}$. We prove that $w(3,k)\ge k^{\Omega (\log k)}$, which perhaps gives some evidence that $k^{\Theta (\log k)}$ is the correct order of magnitude.