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 Erdős–Ko–Rado theorem: when n > 2k, every non-trivial intersecting family of k-subsets of [n] has at most (n−1 k−1 ) − (n−k−1 k−1 ) + 1 members. One extremal family HMn,k consists of a k-set S and all k-subsets of [n] containing a fixed element x 6∈ S and at least one element of S. We prove a degree version of the Hilton–Milner theorem: if n = Ω(k2) and F is a non-trivial intersecting family of k-subsets of [n], then δ(F ) ≤ δ(HMn.k), where δ(F) denotes the minimum (vertex) degree of F . Our proof uses several fundamental results in extremal set theory, the concept of kernels, and a new variant of the Erdős–Ko–Rado theorem.