Unavoidable subgraphs in Ramsey graphs

TL;DR

The paper investigates unavoidable local subgraph structures in Ramsey graphs for a fixed target graph $F$ by leveraging the girth Ramsey theorem to force forest-like arrangements of copies of $F$. It develops both positive forcing results for specific $F$ (notably cycles and balanced complete multipartite graphs) and negative results showing that such forcing can fail for other $F$, including constructions based on ordered hypergraphs. It introduces ordered variants of the girth theorem, derives graphs $F_S$ from hypergraphs, and demonstrates Ramsey graphs that avoid forests of copies of these derived graphs. The work further examines global Ramsey properties via the $2$-density $m_2(F)$, proving that forests do not raise $m_2$ and establishing infinitude results for Ramsey graphs containing cycles, as well as existence results for $F$-free subgraphs of substantial density inside Ramsey graphs. Collectively, the results illuminate the tension between local forest-like structures and global Ramsey phenomena, and they connect combinatorial forcing, hypergraph orderings, and density considerations to map the landscape of Ramsey graphs for a broad class of graphs $F$.

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.

Figures (1)

• Figure 1.1: Some subforests fail to be forests.

Theorems & Definitions (44)

• Definition 1.1: Forests of copies
• Theorem 1.2: Girth Ramsey theorem
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.8
• Theorem 2.1: de Bruijn and Erdős
• Lemma 2.2
• proof