Stability of intersecting families

TL;DR

This paper investigates the stability and maximum size of intersecting families of sets, extending classical theorems by removing size constraints and identifying new extremal configurations.

Contribution

It removes the large n requirement in previous results and fully characterizes the next largest intersecting families beyond the Hilton-Milner type.

Findings

Extended the maximum size results to all n, not just large n.

Identified new extremal families beyond Hilton-Milner.

Answered open questions about the structure of near-maximum intersecting families.

Abstract

The celebrated Erd\H{o}s-Ko-Rado theorem \cite{EKR1961} states that the maximum intersecting $k$-uniform family on $[n]$ is a full star if $n\ge 2k+1$. Furthermore, Hilton-Milner \cite{HM1967} showed that if an intersecting $k$-uniform family on $[n]$ is not a subfamily of a full star, then its maximum size achieves only on a family isomorphic to $HM(n,k):= \Bigl\{G\in {[n] \choose k}: 1\in G, G\cap [2,k+1] \neq \emptyset \Bigr\} \cup \Bigl\{ [2,k+1] \Bigr\} $ if $n>2k$ and $k\ge 4$, and there is one more possibility in the case of $k=3$. Han and Kohayakawa \cite{HK2017} determined the maximum intersecting $k$-uniform family on $[n]$ which is neither a subfamily of a full star nor a subfamily of the extremal family in Hilton-Milner theorm, and they asked what is the next maximum intersecting $k$-uniform family on $[n]$. Kostochka and Mubayi \cite{KM2016} gave the answer for large enough $n$. In this paper, we are going to get rid of the requirement that $n$ is large enough in the result by Kostochka and Mubayi \cite{KM2016} and answer the question of Han and Kohayakawa \cite{HK2017}.