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Forcing Quasirandomness in a Regular Tournament

Jonathan A. Noel, Arjun Ranganathan, Lina M. Simbaqueba

TL;DR

This work investigates when a fixed tournament H forces quasirandomness in sequences of nearly regular tournaments by embedding the problem into tournamenton limit theory. Using a combination of limit-analytic reductions and the flag-algebra method, it derives necessary and sufficient conditions and provides a complete classification of all tournaments on at most 5 vertices that force quasirandomness under near-regularity. Nine five-vertex tournaments do not force, while eleven do, with the latter set including new cases (H_10, H_11, H_13, H_14) proven via flag algebras, in addition to known results. The results illuminate a richer forcing landscape under the near-regular assumption and open questions about infinite families and almost-everywhere forcing phenomena.

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.

Forcing Quasirandomness in a Regular Tournament

TL;DR

This work investigates when a fixed tournament H forces quasirandomness in sequences of nearly regular tournaments by embedding the problem into tournamenton limit theory. Using a combination of limit-analytic reductions and the flag-algebra method, it derives necessary and sufficient conditions and provides a complete classification of all tournaments on at most 5 vertices that force quasirandomness under near-regularity. Nine five-vertex tournaments do not force, while eleven do, with the latter set including new cases (H_10, H_11, H_13, H_14) proven via flag algebras, in addition to known results. The results illuminate a richer forcing landscape under the near-regular assumption and open questions about infinite families and almost-everywhere forcing phenomena.

Abstract

A tournament is said to force quasirandomness if it has the property that a sequence of tournaments of increasing orders is quasirandom if and only if the homomorphism density of in tends to as . 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 are assumed to be "nearly regular." We characterize the tournaments on at most 5 vertices which force quasirandomness under this stronger assumption.
Paper Structure (6 sections, 27 theorems, 162 equations, 1 figure)

This paper contains 6 sections, 27 theorems, 162 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic Reductions
  4. Positive Results
  5. Conclusion
  6. Flag Algebra Coefficients

Key Result

Theorem 1.2

A tournament $H$ forces quasirandomness if and only if it is a transitive tournament on at least 4 vertices or is isomorphic to the tournament $H_{17}$ in Figure fig:smallTourns.

Figures (1)

  • Figure 1: The tournaments on at most $5$ vertices, up to isomorphism. There are 20 such tournaments labelled $H_0,\dots,H_{19}$. The tournaments $H_0,H_1,H_2,H_4$ and $H_8$ are transitive. The tournament $H_3$ is the cyclic tournament on 3 vertices and is also referred to as $C_3$. Similarly, $H_6$ is the unique $4$-vertex tournament containing a spanning cycle and is also referred to as $C_4$.

Theorems & Definitions (51)

  • Definition 1.1
  • Theorem 1.2: See Lovasz79BucicLongShapiraSudakov21CoreglianoParenteSato19Hancock+23CoreglianoRazborov17
  • Definition 1.3
  • Theorem 1.4
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 3.1
  • proof
  • Proposition 3.3
  • proof
  • ...and 41 more