A degree version of the Hilton-Milner theorem

摘要

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 Erdos-Ko-Rado theorem: when n 2k, every non-trivial intersecting family of k-subsets of [n] has at most (n-1k-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 is an element of S and at least one element of S. We prove a degree version of the Hilton-Milner theorem: if n = Omega(k(2)) and F is a non-trivial intersecting family of k-subsets of [n], then delta(F) = (HMn,k), where delta(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 Erdos-Ko-Rado theorem. (C) 2017 Elsevier Inc. All rights reserved.