Forcing Quasirandomness in a Regular Tournament

TL;DR

This paper explores conditions under which certain small tournaments enforce quasirandomness in larger tournaments, especially focusing on nearly regular tournaments and their role in characterizing quasirandomness.

Contribution

It introduces a new variant considering nearly regular tournaments and characterizes small tournaments (up to 5 vertices) that force quasirandomness under this assumption.

Findings

Only one non-transitive tournament forces quasirandomness.

Characterization of small tournaments (up to 5 vertices) that force quasirandomness with nearly regularity.

Contrast with graphs where many such forcing graphs are known.

Abstract

A tournament $H$ is said to force quasirandomness if it has the property that a sequence $(T_n)_{n\in \mathbb{N}}$ of tournaments of increasing orders is quasirandom if and only if the homomorphism density of $H$ in $T_n$ tends to $(1/2)^{\binom{v(H)}{2}}$ as $n\to\infty$. It was recently shown that there is only one non-transitive tournament with this property. This is in contrast to the analogous problem for graphs, where there are numerous graphs that are known to force quasirandomness and the well known Forcing Conjecture suggests that there are many more. To obtain a richer family of characterizations of quasirandomness in tournaments, we propose a variant in which the tournaments $(T_n)_{n\in \mathbb{N}}$ are assumed to be "nearly regular." We characterize the tournaments on at most 5 vertices which force quasirandomness under this stronger assumption.