On vertex Ramsey graphs with forbidden subgraphs

TL;DR

This paper characterizes when certain vertex Ramsey graphs with forbidden subgraphs exist, establishing a necessary and sufficient condition for large color numbers and exploring their occurrence in dense random graphs.

Contribution

It proves the necessity of a classical sufficient condition for the existence of vertex Ramsey graphs with forbidden subgraphs, extending the result to graph families and random graphs.

Findings

Necessary and sufficient condition established for large r

Generalization to graph families

Existence in dense random graphs demonstrated

Abstract

A classical vertex Ramsey result due to Ne\v{s}et\v{r}il and R\"odl states that given a finite family of graphs $\mathcal{F}$, a graph $A$ and a positive integer $r$, if every graph $B\in\mathcal{F}$ has a $2$-vertex-connected subgraph which is not a subgraph of $A$, then there exists an $\mathcal{F}$-free graph which is vertex $r$-Ramsey with respect to $A$. We prove that this sufficient condition for the existence of an $\mathcal{F}$-free graph which is vertex $r$-Ramsey with respect to $A$ is also necessary for large enough number of colours $r$. We further show a generalisation of the result to a family of graphs and the typical existence of such a subgraph in a dense binomial random graph.