Structure and Supersaturation for Intersecting Families

TL;DR

This paper explores supersaturation in intersecting families, analyzing the minimum disjoint pairs in permutations and set families, and characterizes the typical structure of large intersecting families.

Contribution

It introduces a removal lemma for disjoint pairs and determines the structure of extremal families in permutations and set systems.

Findings

Minimum disjoint pairs in large families quantified

Structure of optimal families characterized

Most large intersecting families are trivial for large n

Abstract

The extremal problems regarding the maximum possible size of intersecting families of various combinatorial objects have been extensively studied. In this paper, we investigate supersaturation extensions, which in this context ask for the minimum number of disjoint pairs that must appear in families larger than the extremal threshold. We study the minimum number of disjoint pairs in families of permutations and in $k$-uniform set families, and determine the structure of the optimal families. Our main tool is a removal lemma for disjoint pairs. We also determine the typical structure of $k$-uniform set families without matchings of size $s$ when $n \ge 2sk + 38s^4$, and show that almost all $k$-uniform intersecting families on vertex set $[n]$ are trivial when $n\ge (2+o(1))k$.