A random coloring process gives improved bounds for the Erd\H{o}s-Gy\'arf\'as problem on generalized Ramsey numbers

TL;DR

This paper introduces a randomized coloring process analyzed via differential equations to improve upper bounds on generalized Ramsey numbers, specifically the Erd ext{"o}s-Gyárfás number, for large complete graphs.

Contribution

The paper presents a novel randomized coloring method and its rigorous analysis to achieve better bounds on Erd ext{"o}s-Gyárfás numbers for fixed parameters.

Findings

Improved upper bounds for many fixed p, q values

Application of differential equation method for analysis

Enhanced understanding of coloring processes in Ramsey theory

Abstract

The Erd\H{o}s-Gy\'arf\'as number $f(n, p, q)$ is the smallest number of colors needed to color the edges of the complete graph $K_n$ so that all of its $p$-clique spans at least $q$ colors. In this paper we improve the best known upper bound on $f(n, p, q)$ for many fixed values of $p, q$ and large $n$. Our proof uses a randomized coloring process, which we analyze using the so-called differential equation method to establish dynamic concentration.