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