Constrained Ramsey numbers for rainbow $P_5$

TL;DR

This work investigates the constrained (rainbow) Ramsey number $f(H,P_5)$, the smallest $n$ such that every edge-coloring of the complete graph $K_n$ yields either a monochromatic copy of $H$ or a rainbow copy of $P_5$. The authors develop a general framework showing $f(H,P_5)$ is bounded by 3-color Ramsey numbers in broad classes, and prove $f(H,P_5)=R_3(H)$ for several important families, including disconnected graphs with $\chi(H)\ge 3$, disjoint unions of a connected graph and its subgraphs (e.g., $G_1\subseteq G_2$ implies $f(G_1\cup G_2,P_5)=R_3(G_1\cup G_2)$), and graphs whose components are 3-color-critical or bipartite. They also establish a series of auxiliary results (homological graphs, decomposition families, and structural lemmas) that unify and extend prior results (Gyárfás–Lehel–Schelp; Li et al.) and cover bipartite variants and related questions. The paper highlights both the power and limitations of current structural Ramsey techniques and outlines several open problems, including cases where $R_3(H)<R_2(\mathscr{C}(H))$ and higher-chromatic cases, to guide future work. Overall, the results deepen the connection between constrained Ramsey numbers and classical Ramsey theory, providing a versatile toolkit for rainbow-path problems.

Abstract

Given a graph $H$ and a positive integer $k$, the {\it $k$-colored Ramsey number} $R_k(H)$ is the minimum integer $n$ such that in every $k$-edge-coloring of the complete graph $K_{n}$, there is a monochromatic copy of $H$. Given two graphs $H$ and $G$, the {\it constrained Ramsey number} (also called {\it rainbow Ramsey number}) $f(H,G)$ is defined as the minimum integer $n$ such that, in every edge-coloring of $K_{n}$ with any number of colors, there is either a monochromatic copy of $H$ or a rainbow copy of $G$. Let $P_t$ be the path on $t$ vertices. Gyárfás, Lehel and Schelp proved that $f(H,P_5)=R_3(H)$ when $H$ is a path or a cycle. Li, Besse, Magnant, Wang and Watts conjectured that $f(H,P_5)=R_3(H)$ for any graph $H$, and confirmed this for all connected graphs and all bipartite graphs. In this paper, we address this conjecture for multiple classes of disconnected graphs with chromatic number at least 3. Our newly established general results encompass all known results on this problem. We also obtain several results for a bipartite variation of the problem. In addition, we propose a series of questions concerning this problem from multiple distinct aspects for further research.

Theorems & Definitions (72)

• Theorem 1.1: Erdős-Rado Canonical Ramsey Theorem ErRa
• Conjecture 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Corollary 1.6
• Theorem 1.7
• Theorem 1.8
• Corollary 1.9
• Theorem 1.10