#### Vol. 8, No. 2, 2019

 Recent Issues Volume 8, Issue 2 Volume 8, Issue 1
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A simple proof of the Hilton–Milner theorem

### Peter Frankl

Vol. 8 (2019), No. 2, 97–101
##### Abstract

Let $n\ge 2k\ge 4$ be integers and $\mathsc{ℱ}$ a family of $k$-subsets of $\left\{1,2,\dots ,n\right\}$. We call $\mathsc{ℱ}$ intersecting if $F\cap {F}^{\prime }\ne \varnothing$ for all $F,{F}^{\prime }\in \mathsc{ℱ}$, and we call $\mathsc{ℱ}$ nontrivial if ${\bigcap }_{F\in \mathsc{ℱ}}F=\varnothing$. Strengthening the famous Erdős–Ko–Rado theorem, Hilton and Milner proved that $|\mathsc{ℱ}|\le \left(\genfrac{}{}{0.0pt}{}{n-1}{k-1}\right)-\left(\genfrac{}{}{0.0pt}{}{n-k-1}{k-1}\right)+1$ if $\mathsc{ℱ}$ is nontrivial and intersecting. We provide a proof by injection of this result.

##### Keywords
finite sets, intersection, hypergraphs
Primary: 05D05
##### Milestones
Received: 9 October 2017
Accepted: 28 May 2018
Published: 20 May 2019
##### Authors
 Peter Frankl Rényi Institute Budapest Hungary