Canonical Ramsey numbers of sparse graphs

TL;DR

This paper investigates the Erd ext{"o}s-Rado numbers for sparse graphs, establishing polynomial bounds for bipartite graphs with bounded degree and exploring related constrained Ramsey numbers involving trees and paths.

Contribution

It extends the study of Erd ext{"o}s-Rado numbers to sparse graphs, providing new bounds and analyzing related constrained Ramsey problems involving trees and paths.

Findings

ER(H) is polynomial in |V(H)| for bipartite graphs with bounded degree

ER(H) is exponential in |V(H)| for general sparse graphs

Established nearly optimal bounds for constrained Ramsey numbers involving trees and paths

Abstract

The canonical Ramsey theorem of Erd\H{o}s and Rado implies that for any graph $H$, any edge-coloring (with an arbitrary number of colors) of a sufficiently large complete graph $K_N$ contains a monochromatic, lexicographic, or rainbow copy of $H$. The least such $N$ is called the Erd\H{o}s-Rado number of $H$, denoted by $ER(H)$. Erd\H{o}s-Rado numbers of cliques have received considerable attention, and in this paper we extend this line of research by studying Erd\H{o}s-Rado numbers of sparse graphs. For example, we prove that if $H$ has bounded degree, then $ER(H)$ is polynomial in $|V(H)|$ if $H$ is bipartite, but exponential in general. We also study the closely-related problem of constrained Ramsey numbers. For a given tree $S$ and given path $P_t$, we study the minimum $N$ such that every edge-coloring of $K_N$ contains a monochromatic copy of $S$ or a rainbow copy of $P_t$. We prove a nearly optimal upper bound for this problem, which differs from the best known lower bound by a function of inverse-Ackermann type.