A Random Coloring Process Gives Improved Bounds for the Erdős-Gyárfás Problem on Generalized Ramsey Numbers

The Erdős-Gyárfás 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...

Full description

Saved in:
Bibliographic Details
Published inThe Electronic journal of combinatorics Vol. 32; no. 2
Main Authors Bennett, Patrick, Dudek, Andrzej, English, Sean
Format Journal Article
LanguageEnglish
Published 13.05.2025
Online AccessGet full text

Cover

More Information
Summary:The Erdős-Gyárfás 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.
ISSN:1077-8926
1077-8926
DOI:10.37236/12387