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

TL;DR

This paper establishes the first tight hypergraph analogue of the Andre1sfai--Erdf6s--Sf3s theorem, determining the exact minimum degree threshold for 3-uniform hypergraphs to be 3-partite when free of a generalized triangle.

Contribution

It finds the exact minimum degree condition for hypergraphs to be 3-partite, extending the classic theorem to hypergraphs with optimal bounds.

Findings

Determines the optimal threshold for 3-partiteness in hypergraphs.

Proves the first tight Andre1sfai--Erdf6s--Sf3s type theorem for hypergraphs.

Strengthens previous results on hypergraph structure and cancellative properties.

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. Its extensions to $3$-uniform hypergraphs without the generalized triangle $F_5 = \{abc, abd, cde\}$ have been explored in several previous works such as~\cite{LMR23unif,HLZ24}, demonstrating the existence of $\varepsilon > 0$ such that for large $n$, every $n$-vertex $F_5$-free $3$-graph with minimum degree greater than $(1/9-\varepsilon) n^2$ must be $3$-partite. We determine the optimal value for $\varepsilon$ by showing that for $n \ge 5000$, every $n$-vertex $F_5$-free $3$-graph with minimum degree greater than $4n^2/45$ must be $3$-partite, thus establishing the first tight Andr\'{a}sfai--Erd\H{o}s--S\'{o}s type theorem for hypergraphs. As a corollary, for all positive $n$, every $n$-vertex cancellative $3$-graph with minimum degree greater than $4n^2/45$ must be $3$-partite. This result is also optimal and considerably strengthens prior work, such as that by Bollob\'{a}s~\cite{Bol74} and Keevash--Mubayi~\cite{KM04Cancel}.