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

TL;DR

This paper provides a detailed analysis of the big Ramsey degrees of the universal homogeneous $K_4$-free graph, offering a compact presentation and explicit calculations for small graphs, advancing understanding in combinatorics and graph theory.

Contribution

It offers a self-contained, concise presentation of the big Ramsey degrees for the $K_4$-free graph and computes degrees for small graphs, filling a gap in the existing literature.

Findings

Explicit big Ramsey degrees for small graphs within the $K_4$-free graph.

A compact, self-contained presentation of the case.

Extension of recent characterizations to a specific universal graph.

Abstract

Big Ramsey degrees of Fra\"iss\'e limits of finitely constrained free amalgamation classes in finite binary languages have been recently fully characterised by Balko, Chodounsk\'y, Dobrinen, Hubi\v{c}ka, Kone\v{c}n\'y, 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.