A simple proof of the Hilton–Milner theorem

Abstract

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