On the set-coloring Ramsey numbers of graphs

TL;DR

This paper investigates set-coloring Ramsey numbers for various graphs, providing exact values, bounds, and a lower bound using probabilistic methods, advancing understanding of multi-color graph Ramsey theory.

Contribution

It offers new bounds and exact values for set-coloring Ramsey numbers of specific graphs, and introduces a lower bound via Lovász Local Lemma for general graphs.

Findings

Exact values for stars, paths, matchings, etc.

Bounds for set-coloring Ramsey numbers of various graphs.

A lower bound established using Lovász Local Lemma.

Abstract

The \textit{set-coloring Ramsey number} $\mathrm{R}_{r, s}(G_1,G_2,...,G_r)$ is the least $n \in \mathbb{N}$ such that every coloring $\chi: E\left(K_n\right) \rightarrow\binom{[r]}{s}$ contains a monochromatic copy of $G_i$, that is, a color $i \in[r]$ such that $i \in \chi(e)$ for every $e \in E(G_i)$. If $G_1=G_2=\cdots=G_r=G$, then we write $\mathrm{R}_{r,s}(G)$ for short. In 2022, Le asked to find lower and upper bounds for $\mathrm{R}_{s, t}(G)$ with various kinds of graphs $G$ such as stars, paths, cycles, etc. In this paper, we obtain exact values or bounds for the set-coloring Ramsey numbers of stars, paths, matchings, etc. By Lov\'{a}sz Local Lemma, we give a lower bound for the set-coloring Ramsey number for general graphs.