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

TL;DR

This paper revisits the Erdős–Szemerédi theorem on edge-colorings of complete graphs, clarifying the explicit dependencies between parameters to improve its application in combinatorial arguments.

Contribution

It explicitly determines the parameter dependencies in the Erdős–Szemerédi theorem, enhancing its precision for use in Ramsey-type combinatorial proofs.

Findings

Clarified the dependence between graph size and balance parameter

Improved accuracy of applications in Ramsey theory

Provided explicit bounds for parameters

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.