Improved Lower Bounds for Van Der Waerden Numbers

Abstract

Recently, Ben Green proved that the two-color van der Waerden number ω(3, k) is bounded from below by ${k^{{b_0}\left( k \right)}}$ where ${b_0}\left( k \right) = {c_0}{\left( {{{\log \,k} \over {\log \log \,k}}} \right)^{1/3}}$. We prove a new lower bound of k^b(k) with $b\left( k \right) = {{c\log \,k} \over {\log \log \,k}}$. This is done by modifying Green’s argument, replacing a complicated result about random quadratic forms with an elementary probabilistic result.