On the set-coloring Ramsey numbers of graphs
Mengya He, Yaping Mao
TL;DR
This paper investigates set-coloring Ramsey numbers $\mathrm{R}_{r,s}(G_1,\dots,G_r)$, the least $n$ such that every $(r,s)$-coloring of $E(K_n)$ yields a monochromatic copy of some $G_i$. It combines exact results for specific graph families (notably paths and stars) with general bounds: for paths, tight values such as $\mathrm{R}_{3,2}(P_{3},P_{n},P_{n})=n$ and a broad upper bound $\mathrm{R}_{r,s}(P_{n_1},\dots,P_{n_r})\le\sum_{i=1}^r\frac{n_i-2}{2s}+\frac{1}{2}$, along with a Lovász Local Lemma–based lower bound for general graphs. For stars, the paper derives sharp lower/upper bounds and confirms exact value $\mathrm{R}_{3,2}(K_{1,2},K_{1,n},K_{1,n})=n+1$ under a natural condition. A general lower bound is obtained via the Lovász Local Lemma, and another lower bound is given for families of graphs in terms of their numbers of vertices and edges, linking Ramsey thresholds to extremal graph theory via $\mathrm{ex}(N,H)$. These results advance understanding of how set-color constraints influence Ramsey thresholds and offer tools relevant to coding theory and combinatorial design.
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 $χ: E\left(K_n\right) \rightarrow\binom{[r]}{s}$ contains a monochromatic copy of $G_i$, that is, a color $i \in[r]$ such that $i \in χ(e)$ 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 Lovász Local Lemma, we give a lower bound for the set-coloring Ramsey number for general graphs.