Unavoidable Subtournaments in Tournaments with Large Chromatic Number

TL;DR

This paper investigates the properties of heroic sets of tournaments, especially those with two members, providing necessary conditions and constructing infinitely many minimal heroic sets of size two.

Contribution

It extends the characterization of heroic sets from single tournaments to pairs, offering necessary conditions and explicit constructions for minimal heroic sets.

Findings

Necessary condition for two-tournament heroic sets

Construction of infinitely many minimal heroic sets of size two

Extension of previous single-tournament heroic set characterization

Abstract

For a set H of tournaments, we say H is heroic if every tournament, not containing any member of H as a subtournament, has bounded chromatic number. Berger et al. explicitly characterized all heroic sets containing one tournament. Motivated by this result, we study heroic sets containing two tournaments. We give a necessary condition for a set containing two tournaments to be heroic. We also construct infinitely many minimal heroic sets of size two.