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Extremal problems in uniformly dense hypergraphs and digraphs

Hao Lin, Guanghui Wang, Wenling Zhou, Yiming Zhou

Abstract

The uniform Turán density $π_{u}(F)$ of a $3$-uniform hypergraph (or $3$-graph) $F$ is the supremum of all $d$ such that there exist infinitely many $F$-free $3$-graphs $H$ in which every induced subhypergraph on a linearly sized vertex set has edge density at least $d$. Determining $π_{u}(F)$ for a given $3$-graph $F$ was proposed by Erdős and Sós in the 1980s, yet only a few cases are known. In particular, it remains open whether $1/2$ can occur as a value of $π_{u}$. In this paper, we establish a novel connection between Turán-type extremal problems for digraphs and uniform Turán densities of $3$-graphs. Using digraph extremal results, we give the first verifiable conditions for $3$-graphs $F$ with $π_{u}(F) = (r-1)/r$ and $π_{u}(F) = (r-1)^2/r^2$ for all $r \ge 2$, and identify the corresponding $3$-graphs. In particular, these $3$-graph classes contain some specific $3$-graphs, such as $K^{(3)-}_4$. We also present a sufficient condition ensuring $π_{u}(F)=4/27$ and construct $3$-graphs satisfying it; in particular, our examples are different from the tight $3$-uniform cycles whose uniform Turán density $4/27$ was determined in [{Trans. Amer. Math. Soc. 376 (2023), 4765-4809}]. Finally, we give a short proof of the existence of $3$-graphs $F$ with $π_{u}(F)=1/27$, originally established by Garbe, Král' and Lamaison [{Israel J. Math. 259 (2024), 701-726}] via the hypergraph regularity method.

Extremal problems in uniformly dense hypergraphs and digraphs

Abstract

The uniform Turán density of a -uniform hypergraph (or -graph) is the supremum of all such that there exist infinitely many -free -graphs in which every induced subhypergraph on a linearly sized vertex set has edge density at least . Determining for a given -graph was proposed by Erdős and Sós in the 1980s, yet only a few cases are known. In particular, it remains open whether can occur as a value of . In this paper, we establish a novel connection between Turán-type extremal problems for digraphs and uniform Turán densities of -graphs. Using digraph extremal results, we give the first verifiable conditions for -graphs with and for all , and identify the corresponding -graphs. In particular, these -graph classes contain some specific -graphs, such as . We also present a sufficient condition ensuring and construct -graphs satisfying it; in particular, our examples are different from the tight -uniform cycles whose uniform Turán density was determined in [{Trans. Amer. Math. Soc. 376 (2023), 4765-4809}]. Finally, we give a short proof of the existence of -graphs with , originally established by Garbe, Král' and Lamaison [{Israel J. Math. 259 (2024), 701-726}] via the hypergraph regularity method.
Paper Structure (11 sections, 19 theorems, 83 equations, 2 figures)

This paper contains 11 sections, 19 theorems, 83 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Turán problems in graphs and hypergraphs
  3. Turán problems in uniformly dense hypergraphs
  4. Turán problems in digraphs
  5. Our results
  6. Turán number of the sum of tournaments
  7. Preparation
  8. Proof of \ref{['Thm:copy-three-digraphs']}
  9. Proof of uniform Turán density theorems
  10. Proof of existence theorems
  11. Concluding remarks

Key Result

Theorem 1.2

For every $3$-graph $F$, we have $\pi_{\rm u}(F)=\pi^{\rm pal}_{\rm u}(F)$.

Figures (2)

  • Figure 1: Construction of the proof of \ref{['clm:first pattern']}.
  • Figure 2: Construction of the proof of \ref{['clm:Wi']}.

Theorems & Definitions (47)

  • Definition 1.1
  • Theorem 1.2: Ander
  • Theorem 1.3: brown1970extremal
  • Theorem 1.4: brown1970extremal
  • Definition 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 37 more