The structure of large intersecting families

TL;DR

This paper characterizes the structure of large but not maximum intersecting families of sets, extending classical theorems and providing new stability results using the Delta-system method.

Contribution

It extends the Hilton-Milner theorem, answers a recent question on intersecting families, and describes all such families with more than 10 edges for r=3.

Findings

Extended the Hilton-Milner theorem on nontrivial intersecting families.

Provided a structural description for intersecting families with size below maximum.

Proved a stability result for the Erdős matching problem.

Abstract

A collection of sets is {\em intersecting} if every two members have nonempty intersection. We describe the structure of intersecting families of $r$-sets of an $n$-set whose size is quite a bit smaller than the maximum ${n-1 \choose r-1}$ given by the Erd\H os-Ko-Rado Theorem. In particular, this extends the Hilton-Milner theorem on nontrivial intersecting families and answers a recent question of Han and Kohayakawa for large $n$. In the case $r=3$ we describe the structure of all intersecting families with more than 10 edges. We also prove a stability result for the Erdos matching problem. Our short proofs are simple applications of the Delta-system method introduced and extensively used by Frankl since 1977.