Non-trivial d-wise intersecting families

Abstract

For an integer d≥2, a family F of sets is d-wise intersecting if for any distinct sets A1,A2,…,Ad∈F, A1∩A2∩…∩Ad≠∅, and non-trivial if ⋂A∈FA=∅. Hilton and Milner conjectured that for k≥d≥2 and large enough n, the extremal (i.e. largest) non-trivial d-wise intersecting family of k-element subsets of [n] is, up to isomorphism, one of the following two families:A(k,d)={A∈([n]k):|A∩[d+1]|≥d}H(k,d)={A∈([n]k):[d−1]⊂A,A∩[d,k+1]≠∅}∪{[k+1]∖{i}:i∈[d−1]}. The celebrated Hilton-Milner Theorem states that H(k,2) is the unique, up to isomorphism, extremal non-trivial intersecting family for k>3. We prove the conjecture and prove a stability theorem, stating that any large enough non-trivial d-wise intersecting family of k-element subsets of [n] is a subfamily of A(k,d) or H(k,d).