Characterizing Large Clique Number in Tournaments

TL;DR

This paper proves that large clique numbers in tournaments guarantee the presence of specific large clique subtournaments from two simple families, providing a bounded-size certification and answering a recent open question.

Contribution

It establishes that large clique numbers in tournaments always contain certain large clique subtournaments from two simple families, resolving an open problem.

Findings

Large clique number implies existence of specific large clique subtournaments.

Large clique number is certified by a bounded-size set.

Provides new insights into unavoidable subtournaments in tournaments with large dichromatic number.

Abstract

Aboulker, Aubian, Charbit, and Lopes (2023) defined the clique number of a tournament to be the minimum clique number of one of its backedge graphs. Here we show that if $T$ is a tournament of sufficiently large clique number, then $T$ contains a subtournament of large clique number from one of two simple families of tournaments. In particular, large clique number is always certified by a bounded-size set. This answers a question of Aboulker, Aubian, Charbit, and Lopes (2023), and gives new insight into a line of research initiated by Kim and Kim (2018) into unavoidable subtournaments in tournaments with large dichromatic number.