A sub-exponential transition of the chromatic generalized Ramsey numbers

TL;DR

The paper proves that for fixed q, if a complete graph's edge-coloring with r colors results in unions of q color classes with chromatic number at most 2^q - 1, then the number of vertices n grows sub-exponentially with r, answering a question in generalized Ramsey theory.

Contribution

It establishes a sub-exponential bound on the size of complete graphs under certain coloring constraints, resolving a question posed by Conlon, Fox, Lee, and Sudakov.

Findings

For fixed q, n is sub-exponential in r under the given coloring conditions.

The result provides a new bound in the study of chromatic generalized Ramsey numbers.

Answers an open question in the field of combinatorics and graph theory.

Abstract

A simple graph-product type construction shows that for all natural numbers $r \ge q$, there exists an edge-coloring of the complete graph on $2^r$ vertices using $r$ colors where the graph consisting of the union of arbitrary $q$ color classes has chromatic number $2^q$. We show that for each fixed natural number $q$, if there exists an edge-coloring of the complete graph on $n$ vertices using $r$ colors where the graph consisting of the union of arbitrary $q$ color classes has chromatic number at most $2^q -1 $, then $n$ must be sub-exponential in $r$. This answers a question of Conlon, Fox, Lee, and Sudakov.