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On a hypergraph Mantel theorem

Xizhi Liu

TL;DR

The paper establishes a Mantel-type stability result for $r$-uniform hypergraphs by proving that, for large $n$, the unique extremal $\Delta_r$-free construction is the balanced complete $r$-partite graph $T^{r}(n)$. It combines the entropic density method of $\text{Chao--Yu}$ with the vertex-extendability framework of $\mathrm{LMR}_{23\mathrm{unif}}$ and $\mathrm{HLZ}_{24}$ to obtain a strong stability statement and uniqueness of the extremal structure. The results answer Mubayi–Pikhurko’s question in the large-$n$ regime and extend the classical Mantel/Bollobás-type lines of Turán theory to $r$-graphs, with implications for $\alpha$-spectral Turán problems. The work also clarifies the relationship between Lagrangian density and entropy in the $\mathcal{T}_{r}$-free setting and links the extremal configuration to a familiar, highly structured partitioned graph, highlighting the broader applicability of the entropic approach in hypergraph extremal problems.

Abstract

An $r$-graph is a triangle if there exists a positive integer $i \le \lceil r/2 \rceil$ such that it is isomorphic to the following $r$-graph with three edges: \begin{align*} \left\{\{1, \ldots, r\},~\{1, \ldots, i, r+1, \ldots, 2r-i\},~\{i+1, \ldots, r, r+1, 2r-i+1, \ldots,2r-1\}\right\}. \end{align*} We prove an Andr{á}sfai--Erdős--Sós-type stability theorem for triangle-free $r$-graphs. In particular, it implies that for large $n$, the unique extremal triangle-free construction on $n$ vertices is the balanced complete $r$-partite $r$-graph. The latter result answers a question by Mubayi and Pikhurko~{\cite[Problem~20]{MPS11}} on weakly triangle-free $r$-graphs for large $n$ in a stronger form. The proof combines the recently introduced entropic technique of Chao--Yu~\cite{CY24} with the framework developed in~\cite{LMR23unif,HLZ24}.

On a hypergraph Mantel theorem

TL;DR

The paper establishes a Mantel-type stability result for -uniform hypergraphs by proving that, for large , the unique extremal -free construction is the balanced complete -partite graph . It combines the entropic density method of with the vertex-extendability framework of and to obtain a strong stability statement and uniqueness of the extremal structure. The results answer Mubayi–Pikhurko’s question in the large- regime and extend the classical Mantel/Bollobás-type lines of Turán theory to -graphs, with implications for -spectral Turán problems. The work also clarifies the relationship between Lagrangian density and entropy in the -free setting and links the extremal configuration to a familiar, highly structured partitioned graph, highlighting the broader applicability of the entropic approach in hypergraph extremal problems.

Abstract

An -graph is a triangle if there exists a positive integer such that it is isomorphic to the following -graph with three edges: \begin{align*} \left\{\{1, \ldots, r\},~\{1, \ldots, i, r+1, \ldots, 2r-i\},~\{i+1, \ldots, r, r+1, 2r-i+1, \ldots,2r-1\}\right\}. \end{align*} We prove an Andr{á}sfai--Erdős--Sós-type stability theorem for triangle-free -graphs. In particular, it implies that for large , the unique extremal triangle-free construction on vertices is the balanced complete -partite -graph. The latter result answers a question by Mubayi and Pikhurko~{\cite[Problem~20]{MPS11}} on weakly triangle-free -graphs for large in a stronger form. The proof combines the recently introduced entropic technique of Chao--Yu~\cite{CY24} with the framework developed in~\cite{LMR23unif,HLZ24}.

Paper Structure

This paper contains 9 sections, 11 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic properties of triangle-free hypergraphs
  4. Vertex-extendability
  5. Lagrangian and entropy
  6. Uniqueness of the optimal solution
  7. Vertex-extendability of $\mathbb{T}_{r,1}$
  8. Proof of Theorem \ref{['THM:hypergraph-Mantel']}
  9. Concluding remarks

Key Result

Theorem 1.2

Let $r\ge 2$ be an integer. There exist $\varepsilon = \varepsilon(r) > 0$ and $N_{0} = N_{0}(r)$ such that the following holds for all $n \ge N_0$. Suppose that $\mathcal{H}$ is a $\Delta_{r}$-free $r$-graph on $n$ vertices with $\delta(\mathcal{H}) \ge n^{r-1}/r^{r-1} - \varepsilon n^{r-1}$. Then

Theorems & Definitions (30)

  • Theorem 1.2
  • Theorem 2.3: LMR23unif and HLZ24
  • Lemma 2.5
  • Theorem 2.6: CY24
  • Proposition 2.7
  • proof : Proof of Proposition \ref{['PROP:entropy-difference']}
  • Proposition 3.1
  • Lemma 3.2: CY24
  • Lemma 3.3
  • proof : Proof of Lemma \ref{['LEMMA:entropy-inequlity']}
  • ...and 20 more