On generalized Ramsey numbers in the non-integral regime

TL;DR

This paper extends bounds on generalized Ramsey numbers in the non-integral regime, providing a broader understanding of edge-colorings in graphs, hypergraphs, and list colorings, using the Forbidden Submatching Method.

Contribution

It proves the bound improvement for all non-integral cases and introduces a three-way generalization involving various graph and hypergraph parameters.

Findings

Improved bounds for $f(n,p,q)$ in the non-integral regime.

Generalization to fixed graphs, list coloring, and hypergraphs.

Application of the Forbidden Submatching Method.

Abstract

A $(p,q)$-coloring of a graph $G$ is an edge-coloring of $G$ such that every $p$-clique receives at least $q$ colors. In 1975, Erd\H{o}s and Shelah introduced the generalized Ramsey number $f(n,p,q)$ which is the minimum number of colors needed in a $(p,q)$-coloring of $K_n$. In 1997, Erd\H{o}s and Gy\'arf\'as showed that $f(n,p,q)$ is at most a constant times $n^{\frac{p-2}{\binom{p}{2} - q + 1}}$. Very recently the first author, Dudek, and English improved this bound by a factor of $\log n^{\frac{-1}{\binom{p}{2} - q + 1}} $ for all $q \le \frac{p^2 - 26p + 55}{4}$, and they ask if this improvement could hold for a wider range of $q$. We answer this in the affirmative for the entire non-integral regime, that is, for all integers $p, q$ with $p-2$ not divisible by $\binom{p}{2} - q + 1$. Furthermore, we provide a simultaneous three-way generalization as follows: where $p$-clique is replaced by any fixed graph $F$ (with $|V(F)|-2$ not divisible by $|E(F)| - q + 1$); to list coloring; and to $k$-uniform hypergraphs. Our results are a new application of the Forbidden Submatching Method of the second and fourth authors.