A degree version of the Hilton--Milner theorem

An intersecting family of sets is trivial if all of its members share a common element. Hilton and Milner proved a strong stability result for the celebrated Erd\H{o}s–Ko–Rado theorem: when $n>2k$, every non-trivial intersecting family of $k$-subsets of $[n]$ has at most $\left(\genfrac{}{}{0px}{}{n-1}{k-1}\right)-\left(\genfrac{}{}{0px}{}{n-k-1}{k-1}\right)+1$ members. One extremal family ${\mathcal{H}\mathcal{M}}_{n,k}$ consists of a $k$-set $S$ and all $k$-subsets of $[n]$ containing a fixed element $x\in ̸S$ and at least one element of $S$. We prove a degree version of the Hilton–Milner theorem: if $n=\mathrm{\Omega }(k^2)$ and $\mathcal{F}$ is a non-trivial intersecting family of $k$-subsets of $[n]$, then $\delta (\mathcal{F})\le \delta ({\mathcal{H}\mathcal{M}}_{n\mathrm{.}k})$, where $\delta (\mathcal{F})$ denotes the minimum (vertex) degree of $\mathcal{F}$. Our proof uses several fundamental results in extremal set theory, the concept of kernels, and a new variant of the Erd\H{o}s–Ko–Rado theorem.