Quasirandom forcing in regular tournaments

Date

2025

Authors

Simbaqueba Marin, Lina

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Abstract

The study of quasirandom forcing in various discrete structures has been a well-known problem in Extremal Combinatorics since 1987. In this work, we study quasirandom forcing in the case of tournaments. We say that a tournament H forces quasirandomness if in every quasirandom sequence (T_n)_{n\in \mathbb{N}} of tournaments of increasing order, the density of H in T_n asymptotically equals its expected value. In contrast to the analogous problem in graphs, it was shown that there exists only one non-transitive tournament that forces quasirandomness. To obtain a richer family of tournaments with this property, we propose a variant of it, restricting the definition of quasirandom forcing to only nearly regular sequences of tournaments (T_n)_{n\in \mathbb{N}}. We characterize all tournaments on at most 5 vertices that force quasirandomness under this new setting, obtaining that 11 out of 16 tournaments on four or five vertices are quasirandom forcing in sequences of nearly regular tournaments.

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Keywords

Quasirandomness, Extremal Combinatorics, Tournaments

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