Positive Codegree Andrásfai–Erdős–Sós Theorem for the Generalized Triangle

Abstract

The celebrated Andrásfai–Erdős–Sós 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 $T_r=\{\{1,\dots ,r-1,r\},\{1,\dots ,r-1,r+1\},\{r,r+1,\dots ,2r-1\}\}$: For every $n\ge \left(r-1\right)\left(2r+1\right)/2$, if $\mathscr{H}$ is an $n$-vertex $T_r$-free $r$-uniform hypergraph in which each $\left(r-1\right)$-tuple of vertices is contained in either zero edges or more than $2n/\left(2r+1\right)$ edges of $\mathscr{H}$, then $\mathscr{H}$ is $r$-partite. This result provides the first tight positive codegree Andrásfai–Erdős–Sós type theorem for hypergraphs. It also immediately implies that the positive codegree Turán number of $T_r$ is $\lfloor n/r\rfloor$ for all $r$. Additionally, for $r=3$, our result answers one of the questions posed by Hou et al. in a strong form.