Sidorenko property and forcing in regular tournaments

TL;DR

This paper characterizes which tournaments have the Sidorenko property relative to nearly regular tournaments, identifying those minimized by random tournaments, and explores their forcing properties.

Contribution

It provides a complete characterization of tournaments with the Sidorenko property in the context of nearly regular tournaments, answering open questions about quasirandom forcing.

Findings

Existence of infinitely many non-transitive tournaments that are quasirandom forcing.

Almost every tournament is not quasirandom forcing for nearly regular tournaments.

Complete characterization of Sidorenko property in this setting.

Abstract

We give a complete characterization of tournaments H that have the Sidorenko property with respect to nearly regular tournaments, i.e., the homomorphism density of H among all nearly regular tournaments is minimized by a random tournament. Corollaries of our result are a positive answer to the question of Noel, Ranganathan and Simbaqueba whether there exist infinitely many non-transitive tournaments that are quasirandom forcing for nearly regular tournaments, and a negative answer to their question whether almost every tournament is quasirandom forcing for nearly regular tournaments.