Positive codegree Andr\'{a}sfai--Erd\H{o}s--S\'{o}s theorem for the generalized triangle

TL;DR

This paper extends a classical graph theorem to hypergraphs, establishing conditions under which hypergraphs avoiding a generalized triangle are r-partite, and determines their maximum positive codegree Turán number.

Contribution

It provides the first tight positive codegree Andr{á}sfai--Erd{
51}s--S{ó}s type theorem for hypergraphs, generalizing a fundamental result from graph theory.

Findings

Hypergraphs avoiding the generalized triangle are r-partite under certain codegree conditions.

The positive codegree Turán number of the generalized triangle is exactly  n/r.

Answers a previously open question for the case r=3.

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.