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Towards the Erdős matching conjecture for 4-uniform hypergraphs: stability and applications

Peter Frankl, Hongliang Lu, Jie Ma, Yuze Wu

TL;DR

This work advances the Erdős Matching Conjecture to 4-uniform hypergraphs by proving the EMC for large $n$ in the regime $n\ge 5s$, $e(H) le inom{n}{4}-inom{n-s}{4}$ when $ u(H) le s$. The core technical contribution is a fractional stability result, showing that near-extremal, stable 4-graphs must be close to the canonical extremal $H_{1}$, built from a fractional matching/cover framework. The authors leverage this fractional stability to obtain new exact minimum $d$-degree thresholds for matchings in 5- and 6-uniform hypergraphs, and then push the EMC toolkit to deduce EMC for 4-graphs in a broad range of $n$ and $s$. Overall, the paper introduces a fractional-stability paradigm that unifies extremal counting with fractional covering methods and probabilistic devices to achieve stability and exact matching results in higher uniformities, marking a significant step toward the full EMC for larger $k$.

Abstract

A famous conjecture of Erdős asserts that for $k\ge 3$, the maximum number of edges in an $n$-vertex $k$-uniform hypergraph without $s+1$ pairwise disjoint edges is $\max\{\binom{n}{k}-\binom{n-s}{k},\binom{sk+k-1}{k}\}$. This problem has been central in extremal combinatorics, with substantial progress in the literature, including a complete solution for $k=3$ due to the first author. In this paper, we make progress towards the $4$-uniform case, proving the conjecture for $n\ge 5s$ and sufficiently large $n$, thereby taking a first step analogous to the $3$-uniform case. The main technical contribution is a stability result of independent interest. We further apply this stability to resolve two new instances of conjectures on the minimum $d$-degree threshold for matchings in $5$- and $6$-uniform hypergraphs, in a strengthened form.

Towards the Erdős matching conjecture for 4-uniform hypergraphs: stability and applications

TL;DR

This work advances the Erdős Matching Conjecture to 4-uniform hypergraphs by proving the EMC for large in the regime , when . The core technical contribution is a fractional stability result, showing that near-extremal, stable 4-graphs must be close to the canonical extremal , built from a fractional matching/cover framework. The authors leverage this fractional stability to obtain new exact minimum -degree thresholds for matchings in 5- and 6-uniform hypergraphs, and then push the EMC toolkit to deduce EMC for 4-graphs in a broad range of and . Overall, the paper introduces a fractional-stability paradigm that unifies extremal counting with fractional covering methods and probabilistic devices to achieve stability and exact matching results in higher uniformities, marking a significant step toward the full EMC for larger .

Abstract

A famous conjecture of Erdős asserts that for , the maximum number of edges in an -vertex -uniform hypergraph without pairwise disjoint edges is . This problem has been central in extremal combinatorics, with substantial progress in the literature, including a complete solution for due to the first author. In this paper, we make progress towards the -uniform case, proving the conjecture for and sufficiently large , thereby taking a first step analogous to the -uniform case. The main technical contribution is a stability result of independent interest. We further apply this stability to resolve two new instances of conjectures on the minimum -degree threshold for matchings in - and -uniform hypergraphs, in a strengthened form.
Paper Structure (12 sections, 22 theorems, 69 equations, 1 table)

This paper contains 12 sections, 22 theorems, 69 equations, 1 table.

Table of Contents

  1. Introduction
  2. A fractional stability
  3. Preliminary results
  4. Proof of Theorem \ref{['FStableK4']}
  5. Proof of Theorem \ref{['StableK4']} (Stability)
  6. Proof of Theorem \ref{['EMCK4']}
  7. Proof of Theorem \ref{['VDPC']}
  8. Appendix
  9. Proof of Lemma \ref{['convex']}
  10. Proof of Lemma \ref{['Prebound']}
  11. Proof of Lemma \ref{['Maxvalue']}
  12. Proof of Lemma \ref{['Calculate']}

Key Result

Theorem 1.2

Let $n, s$ be integers such that $n \ge 5s$ and $n \ge n_0$ for some absolute constant $n_0$. Let $H$ be an $n$-vertex $4$-graph with matching number at most $s$. Then $e(H)\leq\binom{n}{4}-\binom{n-s}{4}$.

Theorems & Definitions (48)

  • Conjecture 1.1: Erdős Matching Conjecture (EMC for short), E65
  • Theorem 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5
  • Conjecture 1.6
  • Conjecture 1.7
  • Theorem 1.8
  • Theorem 2.1
  • Proposition 2.2
  • ...and 38 more