Induced subgraphs of induced subgraphs of large chromatic number

TL;DR

This paper demonstrates that for any graph with at least one edge, there exist large chromatic number graphs with controlled induced subgraph properties, extending several recent theorems and revealing strong vertex Ramsey properties.

Contribution

It generalizes recent theorems by establishing the existence of large chromatic graphs with bounded induced subgraph chromatic number for any fixed forbidden subgraph.

Findings

Existence of graphs with arbitrarily large chromatic number and fixed clique number where all F-free induced subgraphs have bounded chromatic number

Extension of Folkman's vertex Ramsey theorem to classes of K_r-free graphs

Results applicable to tournaments, hypergraphs, and graphs with odd girth

Abstract

We prove that, for every graph $F$ with at least one edge, there is a constant $c_F$ such that there are graphs of arbitrarily large chromatic number and the same clique number as $F$ in which every $F$-free induced subgraph has chromatic number at most $c_F$. This generalises recent theorems of Bria\'{n}ski, Davies and Walczak, and Carbonero, Hompe, Moore and Spirkl. Our results imply that for every $r\geq 3$ the class of $K_r$-free graphs has a very strong vertex Ramsey-type property, giving a vast generalisation of a result of Folkman from 1970. We also prove related results for tournaments, hypergraphs and infinite families of graphs, and show an analogous statement for graphs where clique number is replaced by odd girth.