of Birmingham, Moscow Institute of Physics and Technology; Email: [kupavskii@ya.ru]{}.'
author:
- Andrey Kupavskii
title: Structure and properties of large intersecting families
---

Introduction
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For integers $a\le b$, put $[a,b]:=\{a,a+1,\ldots, b\}$, and denote $[n]:=[1,n]$ for shorthand. For a set $X$, denote by $2^{X}$ its power set and, for integer $k\ge 0$, denote by ${X\choose k}$ the collection of all $k$-element subsets ([*$k$-sets*]{}) of $X$. A is simply a collection of sets. We call a family , if any two of its sets intersect. A “trivial” example of an intersecting family is the family of all sets containing a fixed element. We call a family , if the intersection of all sets from the family is empty.

One of the oldest and most famous results in extremal combinatorics is the Erdős–Ko–Rado theorem:

Let $n\ge 2k>0$ and consider an intersecting family ${\mathcal F}\subset {[n]\choose k}$. Then $|{\mathcal F}|\le {n-1\choose k-1}$. Moreover, for $n>2k$ the equality holds only for the families of all $k$-sets containing a given element.

Answering a question of Erdős, Ko, and Rado, Hilton and Milner [@HM] found the size and structure of the largest non-trivial intersecting families of $k$-sets. For $k\ge 4$, up to a permutation of the 