Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Intersecting families with covering number $3$

Andrey Kupavskii

TL;DR

This work resolves the largest intersecting family problem for $k$-subsets of $[n]$ with covering number $3$ in the regime $k \\ge 100$ and $n > 2k$, showing the extremal family is the same $\,\mathcal{C}_3(n,k)$ identified by Frankl, built from the $T_2(k)$-based structure. The authors develop two complementary methods: peeling with spread approximations to handle large diversity $\\gamma(\\mathcal{F})$, and a bipartite switching technique to force a tau-2 core within $\\mathcal{F}(\\bar{1})$ that must be isomorphic to $\\mathcal{T}_2(k)$ in the optimal case. Together these yield sharp size bounds and structural characterization, establishing that the same extremal family governs tau=3 in essentially all large-parameter scenarios. The results advance extremal set theory by providing a flexible toolkit—bipartite switching and spread/peeling—that may extend to tau \\ge 4$ cases, and clarifying the landscape beyond the classical Erdős–Ko–Rado regime.

Abstract

A covering number of a family is the size of the smallest set that intersects all sets from the family. In 1978 Frankl determined for $n\ge n_0(k)$ the largest intersecting family of $k$-element subsets of $[n]$ with covering number $3$. In this paper, we essentially settle this problem, showing that the same family is extremal for any $k\ge 100$ and $n>2k$.

Intersecting families with covering number $3$

TL;DR

This work resolves the largest intersecting family problem for -subsets of with covering number in the regime and , showing the extremal family is the same identified by Frankl, built from the -based structure. The authors develop two complementary methods: peeling with spread approximations to handle large diversity , and a bipartite switching technique to force a tau-2 core within that must be isomorphic to in the optimal case. Together these yield sharp size bounds and structural characterization, establishing that the same extremal family governs tau=3 in essentially all large-parameter scenarios. The results advance extremal set theory by providing a flexible toolkit—bipartite switching and spread/peeling—that may extend to tau \\ge 4$ cases, and clarifying the landscape beyond the classical Erdős–Ko–Rado regime.

Abstract

A covering number of a family is the size of the smallest set that intersects all sets from the family. In 1978 Frankl determined for the largest intersecting family of -element subsets of with covering number . In this paper, we essentially settle this problem, showing that the same family is extremal for any and .
Paper Structure (7 sections, 8 theorems, 41 equations)

This paper contains 7 sections, 8 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thmtau3']}
  3. Outline of the approach
  4. Preliminaries
  5. Proof of Theorem \ref{['thmtau3']}
  6. The case $\gamma(\mathcal{F})>{n-5\choose k-3}$
  7. The case $\gamma(\mathcal{F})\le {n-5\choose k-3}$

Key Result

Theorem 1.1

Let $n>2k>0$ and $\mathcal{F}\subset {[n]\choose k}$ be an intersecting family. If $\gamma(\mathcal{F})\ge {n-u-1\choose n-k-1}$ for some real $3\le u\le k$, then

Theorems & Definitions (11)

  • Theorem 1.1: KZ
  • Theorem 1.2: F16
  • Theorem 1.3
  • Corollary 2.1: KZ
  • Lemma 2.2
  • proof : Proof of Lemma \ref{['lemmin']}
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Lemma 2.5
  • ...and 1 more