Seokbeom Kim (김석범), The structure of △(1, 2, 2)-free tournaments

Given a tournament $S$, a tournament is $S$-free if it has no subtournament isomorphic to $S$. Until now, there have been only a small number of tournaments $S$ such that the complete structure of $S$-free tournaments is known.

Let $\mathrm{△}(1,2,2)$ be a tournament obtained from the cyclic triangle by substituting two-vertex tournaments for two of its vertices. In this talk, we present a structure theorem for $\mathrm{△}(1,2,2)$-free tournaments, which was previously unknown. As an application, we provide tight bounds for the chromatic number as well as the size of the largest transitive subtournament for such tournaments.

This talk is based on joint work with Taite LaGrange, Mathieu Rundström, Arpan Sadhukhan, and Sophie Spirkl.