A Hilton–Milner-type theorem and an intersection conjecture for signed sets

Abstract

A family A of sets is said to be intersecting if any two sets in A intersect (i.e. have at least one common element). A is said to be centred if there is an element common to all the sets in A; otherwise, A is said to be non-centred. For any r ∈ [n] := {1, . . . , n} and any integer k ≥ 2, let Sn,r,k be the family {{(x1, y1), . . . , (xr, yr)}: x1, . . . , xr are distinct elements of [n], y1, . . . , yr ∈ [k]} of k-signed r-sets on [n]. Let m := max{0, 2r−n}.We establish the following Hilton–Milner-type theorems, the second of which is proved using the first: (i) If A1 and A2 are non-empty cross-intersecting (i.e. any set in A1 intersects any set in A2) sub-families of Sn,r,k, then |A1| + |A2| ≤ n R K r −r i=m r I (k − 1) I n – r r – I K r−i + 1. (ii) If A is a non-centred intersecting sub-family of Sn,r,k, 2 ≤ r ≤ n, then |A| ≤ n – 1 r – 1 K r−1 −r−1 i=m r I (k − 1) I n − 1 – r r − 1 – I K r−1−i + 1 if r < n; k r−1 − (k − 1) r−1 + k − 1 if r = n. We also determine the extremal structures. (ii) is a stability theorem that extends Erdős–Ko–Rado-type results proved by various authors. We then show that (ii) leads to further evidence for an intersection conjecture suggested by the author about general signed set systems.