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On resilient hypergraphs

Peter Frankl, Jian Wang

TL;DR

This paper studies resilience of $k$-uniform hypergraphs under vertex deletions that preserve the matching number, focusing on the case $k=3$ and $s=2$. The authors develop structural tools around the $(k-1)$-resilient framework, leveraging the $\mathcal{R}=\mathcal{T}^s$ construction and König–Hall arguments to enforce strong intersection properties, complemented by weight-based sunflower counts. They establish the exact maximum size $m(3,2)=\binom{8}{3}=56$ for 2-resilient 3-graphs with $\nu=2$ and provide a substantial improvement on upper bounds for $s\ge 3$ (Theorem 1.9). The work also analyzes both the $3$-intersecting and not-$3$-intersecting$\,$$\mathcal{R}$ regimes and ends with conjectures guiding future exploration of general $k$ and $s$ values.

Abstract

The matching number of a $k$-graph is the maximum number of pairwise disjoint edges in it. The $k$-graph is called $t$-resilient if omitting $t$ vertices never decreases its matching number. The complete $k$-graph on $sk+k-1$ vertices has matching number $s$ and it is easily seen to be $(k-1)$-resilient. We conjecture that this is maximal for $k=3$ and $s$ arbitrary. The main result verifies this conjecture for $s=2$. Then Theorem 1.9 provides a considerable improvement on the known upper bounds for $s\geq 3$.

On resilient hypergraphs

TL;DR

This paper studies resilience of -uniform hypergraphs under vertex deletions that preserve the matching number, focusing on the case and . The authors develop structural tools around the -resilient framework, leveraging the construction and König–Hall arguments to enforce strong intersection properties, complemented by weight-based sunflower counts. They establish the exact maximum size for 2-resilient 3-graphs with and provide a substantial improvement on upper bounds for (Theorem 1.9). The work also analyzes both the -intersecting and not--intersecting regimes and ends with conjectures guiding future exploration of general and values.

Abstract

The matching number of a -graph is the maximum number of pairwise disjoint edges in it. The -graph is called -resilient if omitting vertices never decreases its matching number. The complete -graph on vertices has matching number and it is easily seen to be -resilient. We conjecture that this is maximal for and arbitrary. The main result verifies this conjecture for . Then Theorem 1.9 provides a considerable improvement on the known upper bounds for .

Paper Structure

This paper contains 7 sections, 22 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. Some simple facts and the proof of Theorem \ref{['thm-main3']}
  3. Two special cases of Theorem \ref{['thm-main']}
  4. The proof of Theorem \ref{['thm-main']} in the case $\mathcal{R}=\mathcal{T}^2$ is 3-intersecting
  5. Proof of Lemma \ref{['lem-main']}
  6. The case $\mathcal{R}$ is not 3-intersecting
  7. Concluding remarks

Key Result

Theorem 1.2

Let $\mathcal{F}\subset \binom{[n]}{3}$ be a family with matching number $s$. If $n\geq 3(s+1)$, then

Theorems & Definitions (60)

  • Conjecture 1.1: Erdős Matching Conjecture E65
  • Theorem 1.2: F12
  • Definition 1.3
  • Conjecture 1.4
  • Definition 1.5
  • Theorem 1.6
  • Lemma 1.7
  • Theorem 1.8: lovasz
  • Theorem 1.9
  • proof
  • ...and 50 more