The chromatic number of a tournament T is the smallest number of
transitive tournaments that partition V(T). Let us say that a tournament
S is a hero if for every tournament T not containing S, the chromatic
number of T is at most a constant c(S). Recently, in joint work with Eli
Berger, Krzysztof Choromanski, Jacob Fox, Martin Loebl, Alex Scott, Paul
Seymour and Stephan Thomasse, we proved a theorem that gives a complete
description of all heroes. This talk will describe the result, and
survey some of the proof ideas.