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Sidorenko property and forcing in regular tournaments

Daniel Král', Matjaž Krnc, Filip Kučerák, Bernard Lidický, Jan Volec

TL;DR

This work characterizes which tournaments H have the Sidorenko property with respect to nearly regular tournaments, showing that exactly transitive tournaments and blow-ups T[a,b,c] of the cyclically oriented triangle with each part transitive minimize the homomorphism density against regular tournamentons. The authors develop a limit-analytic framework using tournamentons, establish key density inequalities for blow-ups C[a,b,c] via auxiliary functions and standard inequalities, and prove that equality occurs only for the uniform (random) tournamenton W≡1/2. They also prove a witness construction for all other H, demonstrating the existence of regular tournamentons with t(H,W)=0, and derive forcing consequences, including infinite families of non-transitive forcing tournaments and a negative answer to whether almost every tournament is forcing in this setting. Overall, the paper advances our understanding of Sidorenko-type phenomena and quasirandom forcing in tournaments, clarifying which structures drive minimization of subgraph densities in nearly regular regimes and outlining remaining open questions, especially for more complex blow-ups and entropy-based proofs.

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.

Sidorenko property and forcing in regular tournaments

TL;DR

This work characterizes which tournaments H have the Sidorenko property with respect to nearly regular tournaments, showing that exactly transitive tournaments and blow-ups T[a,b,c] of the cyclically oriented triangle with each part transitive minimize the homomorphism density against regular tournamentons. The authors develop a limit-analytic framework using tournamentons, establish key density inequalities for blow-ups C[a,b,c] via auxiliary functions and standard inequalities, and prove that equality occurs only for the uniform (random) tournamenton W≡1/2. They also prove a witness construction for all other H, demonstrating the existence of regular tournamentons with t(H,W)=0, and derive forcing consequences, including infinite families of non-transitive forcing tournaments and a negative answer to whether almost every tournament is forcing in this setting. Overall, the paper advances our understanding of Sidorenko-type phenomena and quasirandom forcing in tournaments, clarifying which structures drive minimization of subgraph densities in nearly regular regimes and outlining remaining open questions, especially for more complex blow-ups and entropy-based proofs.

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.
Paper Structure (6 sections, 12 theorems, 61 equations, 9 figures)

This paper contains 6 sections, 12 theorems, 61 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Sketch of the main argument
  3. Preliminaries
  4. Constructions
  5. Main result
  6. Conclusion

Key Result

Proposition 1

Let $n\in{\mathbb N}$ and let $T$ be the $n$-vertex transitive tournament. For every tournamenton $W$, it holds that Moreover, the equality holds if and only if

Figures (9)

  • Figure 1: The unique quasirandom forcing tournament that is not transitive.
  • Figure 2: The digraphs $S^+[3]$, $S^+[4]$, $S^-[3]$ and $S^-[4]$.
  • Figure 3: The digraphs $C[2,1,1]$, $C[3,1,1]$ and $C[2,2,1]$, which are blow-ups of the cyclically oriented triangle with parts of the sizes given by the parameters.
  • Figure 4: The tournaments $T[2,1,1]$, $T[3,1,1]$ and $T[2,2,1]$.
  • Figure 5: The digraphs $B[2]$, $B[3]$ and $B[4]$.
  • ...and 4 more figures

Theorems & Definitions (22)

  • Proposition 1
  • Proposition 2
  • Lemma 3
  • proof
  • Theorem 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • ...and 12 more