Bounded VC-dimension implies the Schur-Erdos conjecture

Abstract

In 1916, Schur introduced the Ramsey number $r(3;m)$, which is the minimum integer $n$ such that for any $m$-coloring of the edges of the complete graph $K_n$, there is a monochromatic copy of $K_3$. He showed that $r(3;m) \leq O(m!)$, and a simple construction demonstrates that $r(3;m) \geq 2^{\Omega(m)}$. An old conjecture of Erd\H os states that $r(3;m) = 2^{\Theta(m)}$. In this note, we prove the conjecture for $m$-colorings with bounded VC-dimension, that is, for $m$-colorings with the property that the set system $\mathcal{F}$ induced by the neighborhoods of the vertices with respect to each color class has bounded VC-dimension.