Positive codegree Andrásfai--Erdős--Sós theorem for the generalized triangle

Abstract: The celebrated Andr\'{a}sfai--Erd\H{o}s--S\'{o}s Theorem from 1974 shows that every $n$-vertex triangle-free graph with minimum degree greater than $2n/5$ must be bipartite. We establish a positive codegree extension of this result for the $r$-uniform generalized triangle $\mathrm{T}{r} = \left{{1,\ldots, r-1,r}, {1,\ldots, r-1,r+1},{r,r+1, \ldots, 2r-1}\right}$$\colon$ For every $n \ge (r-1)(2r+1)/2$, if $\mathcal{H}$ is an $n$-vertex $\mathrm{T}{r}$-free $r$-uniform hypergraph in which each $(r-1)$-tuple of vertices is contained in either zero edges or more than $2n/(2r+1)$ edges of $\mathcal{H}$, then $\mathcal{H}$ is $r$-partite. This result provides the first tight positive codegree Andr{\'a}sfai--Erd\H{o}s--S\'{o}s type theorem for hypergraphs. It also immediately implies that the positive codegree Tur\'{a}n number of $\mathrm{T}_{r}$ is $\lfloor n/r \rfloor$ for all $r$. Additionally, for $r=3$, our result answers one of the questions posed by Hou et al.~\cite{HLYZZ22} in a strong form.