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

Xizhi Liu, Sijie Ren, Jian Wang

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. 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 Andrásfai--Erdős--Sós 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 Bollobás~\cite{Bol74} and Keevash--Mubayi~\cite{KM04Cancel}.

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

Abstract

The celebrated Andrásfai--Erdős--Sós Theorem from 1974 shows that every -vertex triangle-free graph with minimum degree greater than must be bipartite. Its extensions to -uniform hypergraphs without the generalized triangle have been explored in several previous works such as~\cite{LMR23unif,HLZ24}, demonstrating the existence of such that for large , every -vertex -free -graph with minimum degree greater than must be -partite. We determine the optimal value for by showing that for , every -vertex -free -graph with minimum degree greater than must be -partite, thus establishing the first tight Andrásfai--Erdős--Sós type theorem for hypergraphs. As a corollary, for all positive , every -vertex cancellative -graph with minimum degree greater than must be -partite. This result is also optimal and considerably strengthens prior work, such as that by Bollobás~\cite{Bol74} and Keevash--Mubayi~\cite{KM04Cancel}.

Paper Structure

This paper contains 12 sections, 17 theorems, 86 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graphs
  4. Hypergraphs
  5. Proofs of Theorems \ref{['THM:main-AES-F5']} and \ref{['THM:main-AES-K43-F5']}
  6. Proof of Proposition \ref{['PROP:F5-shadow-no-K4']}
  7. Preparations for the proof of Proposition \ref{['PROP:AES-shadow-K4']}
  8. Proof of Lemma \ref{['LEMMA:U1-U2-U3-3-partite']}
  9. Proof of Lemma \ref{['LEMMA:opt-1']}
  10. Proof of Lemma \ref{['LEMMA:opt-2']}
  11. Proof of Proposition \ref{['PROP:AES-shadow-K4']}
  12. Concluding remarks

Key Result

Theorem 1.1

For $n \ge 5000$, every $n$-vertex $F_5$-free $3$-graph with $\delta(\mathcal{H}) > \frac{4n^2}{45}$ is $3$-partite.

Figures (2)

  • Figure 1: The $3$-uniform $5$-wheel $W_{5}^{3}$.
  • Figure 2: The graphs $\Gamma_1, \Gamma_2, \Gamma_3, \Gamma_4, \Gamma_5$.

Theorems & Definitions (44)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1: Lyl14
  • Theorem 2.2: Jin93
  • Theorem 2.3: Moon68
  • Theorem 2.4: MM62
  • Proposition 3.1
  • Proposition 3.2
  • proof : Proof of Theorem \ref{['THM:main-AES-F5']}
  • proof : Proof of Theorem \ref{['THM:main-AES-K43-F5']}
  • ...and 34 more