A degree version of the Hilton–Milner theorem

Abstract

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−1k−1)−(n−k−1k−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∉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.