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A note on hypergraph extensions of Mantel's theorem

Jie Ma, Tianming Zhu

TL;DR

The paper sharpens the entropy-based approach to hypergraph Turán problems in the Mantel-type regime by providing a shorter, optimally bounded proof that the density of the family $\mathcal{F}_k^{\le t}$ equals $k!/k^k$, where $t=t(k)$ is defined by $\sum_{i=t}^k \frac{1}{i}>1$ (with $t(k)$ at most on the order of $k/e$). It builds on the Chao-Yu framework, leveraging ratio-sequence inequalities and a novel combinatorial construction to bound the product of ratio terms, yielding the upper bound, while a complete $k$-partite construction furnishes the matching lower bound. The authors extend the result to a broader class of $\lambda$-tents $\Delta_\lambda$ under a subset-sum condition, via homomorphism arguments, thereby unifying and slightly strengthening previous results by Il'kovič and Yan. This refinement advances the understanding of hypergraph Turán densities and highlights the power and limits of entropy methods in extremal combinatorics.

Abstract

Chao and Yu introduced an entropy method for hypergraph Turán problems, and used it to show that the family of $\lfloor k/2\rfloor$ $k$-uniform tents have Turán density $k!/k^k$. Il'kovič and Yan improved this by reducing to a subfamily of $\lceil k/e\rceil$ tents. In this note, enhancing Il'kovič-Yan's result, we give a significantly shorter entropy proof, with optimal bounds within this framework.

A note on hypergraph extensions of Mantel's theorem

TL;DR

The paper sharpens the entropy-based approach to hypergraph Turán problems in the Mantel-type regime by providing a shorter, optimally bounded proof that the density of the family equals , where is defined by (with at most on the order of ). It builds on the Chao-Yu framework, leveraging ratio-sequence inequalities and a novel combinatorial construction to bound the product of ratio terms, yielding the upper bound, while a complete -partite construction furnishes the matching lower bound. The authors extend the result to a broader class of -tents under a subset-sum condition, via homomorphism arguments, thereby unifying and slightly strengthening previous results by Il'kovič and Yan. This refinement advances the understanding of hypergraph Turán densities and highlights the power and limits of entropy methods in extremal combinatorics.

Abstract

Chao and Yu introduced an entropy method for hypergraph Turán problems, and used it to show that the family of -uniform tents have Turán density . Il'kovič and Yan improved this by reducing to a subfamily of tents. In this note, enhancing Il'kovič-Yan's result, we give a significantly shorter entropy proof, with optimal bounds within this framework.
Paper Structure (4 sections, 7 theorems, 10 equations, 1 table)

This paper contains 4 sections, 7 theorems, 10 equations, 1 table.

Table of Contents

  1. Introduction
  2. The entropy method of Chao-Yu
  3. Proof of the results
  4. Proof of Lemma \ref{['LEMMA:Final-Inequality']}

Key Result

Theorem 1.1

For every integer $k \geq 2$, $\pi(\mathcal{F}_k^{\le t(k)}) = k!/k^k.$

Theorems & Definitions (11)

  • Theorem 1.1
  • Corollary 1.2
  • Lemma 2.1: Chao-Yu, chao2024entropymeetsturannew
  • Lemma 2.2: Chao-Yu, chao2024entropymeetsturannew
  • Lemma 2.3
  • Lemma 3.1
  • proof : Proof of Theorem \ref{['THM:Main']}, using Lemma \ref{['LEMMA:Final-Inequality']}.
  • proof : Proof of Corollary \ref{['Coro:Main-2']}.
  • Lemma 3.2
  • proof
  • ...and 1 more