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

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.

6 pages; extended abstract accepted for Eurocomb2025; minor revision with additional big Ramsey degrees computed