On a hypergraph Mantel theorem

TL;DR

This paper establishes a stability theorem for triangle-free hypergraphs, showing that large extremal structures are balanced complete r-partite hypergraphs, thus answering a significant open question in hypergraph theory.

Contribution

It proves a stability theorem for triangle-free r-graphs, identifying the unique extremal configuration as the balanced complete r-partite hypergraph for large n.

Findings

Proves a stability theorem for triangle-free r-graphs.

Identifies the extremal structure as the balanced complete r-partite hypergraph.

Answers a question by Mubayi and Pikhurko on weakly triangle-free r-graphs.

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{\'a}sfai--Erd\H{o}s--S\'{o}s-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}.