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Intersecting families of sets are typically trivial

József Balogh, Ramon I. Garcia, Lina Li, Adam Zsolt Wagner

Abstract

A family of subsets of $[n]$ is intersecting if every pair of its sets intersects. Determining the structure of large intersecting families is a central problem in extremal combinatorics. Frankl-Kupavskii and Balogh-Das-Liu-Sharifzadeh-Tran independently showed that for $n\geq 2k + c\sqrt{k\ln k}$, almost all $k$-uniform intersecting families are stars. Improving their result, we show that the same conclusion holds for $n\geq 2k+ 100\ln k$. Our proof uses, among others, Sapozhenko's graph container lemma and the Das-Tran removal lemma.

Intersecting families of sets are typically trivial

Abstract

A family of subsets of is intersecting if every pair of its sets intersects. Determining the structure of large intersecting families is a central problem in extremal combinatorics. Frankl-Kupavskii and Balogh-Das-Liu-Sharifzadeh-Tran independently showed that for , almost all -uniform intersecting families are stars. Improving their result, we show that the same conclusion holds for . Our proof uses, among others, Sapozhenko's graph container lemma and the Das-Tran removal lemma.

Paper Structure

This paper contains 11 sections, 17 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Preliminary
  4. Relation to independent sets
  5. Tools
  6. The graph container theorem
  7. Outline of the proof of Theorem 1.3
  8. Intersecting families with large components
  9. Independent sets with small components
  10. Proof of Theorem \ref{['containerlemmalayer']}
  11. Graph container lemma for biregular graphs

Key Result

Theorem 1.1

For $n\geq 2k +2+c\sqrt{k \ln k}$ and $k \rightarrow\infty$ we have where $c=2$ in frankl2018counting and $c$ was a large constant in balogh2018structure.

Theorems & Definitions (34)

  • Theorem 1.1: Balogh--Das--Liu--Sharifzadeh--Tran balogh2018structure, Frankl--Kupavskii frankl2018counting
  • Theorem 1.2: Balogh--Das--Delcourt--Liu--Sharifzadeh balogh2015intersecting
  • Theorem 1.3
  • Conjecture 1.4
  • Conjecture 1.5
  • Proposition 2.1
  • Lemma 2.2
  • Corollary 2.3
  • Theorem 2.4: Lovász lovasz2007combinatorial
  • Theorem 2.5: Isoperimetry
  • ...and 24 more