Let
$n\ge 2k\ge 4$ be integers
and
$\mathcal{\mathcal{F}}$ a family
of
$k$subsets
of
$\left\{1,2,\dots ,n\right\}$. We call
$\mathcal{\mathcal{F}}$
intersecting
if
$F\cap {F}^{\prime}\ne \varnothing $ for all
$F,{F}^{\prime}\in \mathcal{\mathcal{F}}$, and we call
$\mathcal{\mathcal{F}}$
nontrivial
if
${\bigcap}_{F\in \mathcal{\mathcal{F}}}F=\varnothing $.
Strengthening the famous Erdős–Ko–Rado theorem, Hilton and Milner proved that
$\left\mathcal{\mathcal{F}}\right\le \left(\genfrac{}{}{0.0pt}{}{n1}{k1}\right)\left(\genfrac{}{}{0.0pt}{}{nk1}{k1}\right)+1$ if
$\mathcal{\mathcal{F}}$ is
nontrivial and intersecting. We provide a proof by injection of this result.
