Induced Ramsey problems for trees and graphs with bounded treewidth

TL;DR

This paper proves that for graphs with bounded degree and treewidth, the induced size-Ramsey number grows linearly with the number of vertices, extending classical results in Ramsey theory.

Contribution

It establishes a linear bound on the induced Ramsey number for graphs with bounded degree and treewidth, using a novel reduction approach.

Findings

Linear bound on induced Ramsey numbers for bounded degree and treewidth graphs

Extension of classical Ramsey theory results to induced subgraphs

Simple proof technique based on a new reduction argument

Abstract

The induced $q$-color size-Ramsey number $\hat{r}_{\text{ind}}(H;q)$ of a graph $H$ is the minimal number of edges a host graph $G$ can have so that every $q$-edge-coloring of $G$ contains a monochromatic copy of $H$ which is an induced subgraph of $G$. A natural question, which in the non-induced case has a very long history, asks which families of graphs $H$ have induced Ramsey numbers that are linear in $|H|$. We prove that for every $k,w,q$, if $H$ is an $n$-vertex graph with maximum degree $k$ and treewidth at most $w$, then $\hat{r}_{\text{ind}}(H;q) = O_{k,w,q}(n)$. This extends several old and recent results in Ramsey theory. Our proof is quite simple and relies upon a novel reduction argument.