Some results and problems on tournament structure

Abstract

This paper is a survey of results and problems related to the following question: is it true that if G is a tournament with sufficiently large chromatic number, then G has two vertex-disjoint subtournaments $A,B$, both with large chromatic number, such that all edges between them are directed from A to B? We describe what we know about this question, and report some progress on several other related questions, on tournament colouring and domination.