Remarks on a theorem of Erd\H{o}s and Szemer\'{e}di

Abstract

Given a graph $G$ and a real $\varepsilon>0$, an edge-coloring of $G$ is called $\varepsilon$-balanced if each color appears on at least an $\varepsilon$-fraction of the edges in $G$. A classical result of Erd\H{o}s and Szemer\'{e}di asserts that if a $2$-edge-coloring of a complete graph $K_n$ is not $\varepsilon$-balanced for some $0<\varepsilon\leq1/2$, then there exists a large monochromatic clique. This theorem has been used extensively in Ramsey-type arguments, as it allows one to focus on reasonably balanced colorings. However, in its original formulation the dependence between $n$ and $\varepsilon$ was left implicit, occasionally leading to inaccurate applications. In this short note, we revisit the Erd\H{o}s--Szemer\'{e}di theorem and specify all parameter dependencies.