Unavoidable subgraphs in Ramsey graphs

TL;DR

This paper investigates the structure of large Ramsey graphs, showing that they contain certain subgraphs called forests of copies of a given graph, and explores the extent of their presence.

Contribution

It introduces new structural insights into Ramsey graphs, demonstrating the unavoidable presence of specific subgraph configurations.

Findings

Ramsey graphs contain forests of copies of the given graph F

Structural properties of these forests are characterized

Implications for the existence of certain subgraphs in large Ramsey graphs

Abstract

We study subgraphs that appear in large Ramsey graphs for a given graph $F$. The recent girth Ramsey theorem of the first two authors asserts that there are Ramsey graphs such that all small subgraphs are `forests of copies of $F$' amalgamated on vertices and edges. We derive a few further consequences from this structural result and investigate to which extent such forests of copies must be present in Ramsey graphs.