Remarks on a theorem of Erdős and Szemerédi
Dingyuan Liu
TL;DR
This note clarifies the parameter dependencies in the Erdős–Szemerédi theorem for unbalanced $2$-edge-colorings by presenting a precise formulation: for $0<C\le0.01$, $n\ge3$, and $2\le k\le n/(3C)$, any $n$-vertex graph with at least $(1-1/k)\binom{n}{2}$ edges contains a clique or independent set of size at least $\frac{Ck\log n}{\log k}$. It then provides a rigorous proof via a three-case analysis ($k\le100$, $k\ge\sqrt{n}$, and $100<k<\sqrt{n}$), employing Ramsey-number bounds, Turán's theorem, and a structural refinement of the Erdős–Szemerédi argument to achieve the explicit bound. The result ensures precise finite-n applicability for Ramsey-type arguments and eliminates ambiguity in parameter relationships between $n$, $k$, and the guaranteed homogeneous structures. Overall, the work tightens the theorem's use in combinatorial proofs by making the dependencies explicit and essentially optimal.
Abstract
Given a graph $G$ and a real $\varepsilon>0$, an edge-coloring of $G$ is called \textit{$\varepsilon$-balanced} if each color appears on at least an $\varepsilon$-fraction of the edges in $G$. A classical result of Erdős and Szemerédi 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 Erdős--Szemerédi theorem and specify all parameter dependencies.