Counting big Ramsey degrees of the homogeneous and universal $K_4$-free graph

TL;DR

Addresses the problem of determining the exact big Ramsey degrees of finite subgraphs in the countable universal homogeneous $K_4$-free graph $\mathbf{R}_4$. Adopts a self-contained diary-based approach tailored to $\mathbf{R}_4$, including a compact presentation of the seven diary update rules and the encoding of vertex-types by words over $\Sigma=\{0,1,2\}$. Proves that for every finite $\mathbf{K}_4$-free graph $\mathbf{G}$, the big Ramsey degree in $\mathbf{R}_4$ equals $|T(\mathbf{G})|\cdot |\mathrm{Aut}(\mathbf{G})|$, where $T(\mathbf{G})$ is the set of $\mathbf{K}_4$-free diaries with $\mathbf{G}(S)\cong \mathbf{G}$. Characterizes how gadgets realizing a meet or an age-change control closures, and provides groundwork for computing big Ramsey degrees for small graphs within this natural homogeneous, clique-free setting.

Abstract

Big Ramsey degrees of Fraïssé limits of finitely constrained free amalgamation classes in finite binary languages have been recently fully characterised by Balko, Chodounský, Dobrinen, Hubička, Konečný, Vena, and Zucker. A special case of this characterisation is the universal homogeneous $K_4$-free graph. We give a self-contained and relatively compact presentation of this case and compute the actual big Ramsey degrees of small graphs.

Theorems & Definitions (2)

• Definition 2.1: $\mathbf {K}_4$-free diaries
• Theorem 2.2