On a hypergraph Mantel theorem

An $r$-graph is a triangle if there exists a positive integer $i\le \lceil r\mathrm{/}2\rceil$ 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}.