Probably Intersecting Families are Not Nested

Abstract

It is well known that an intersecting family of subsets of an n-element set can contain at most 2n−1 sets. It is natural to wonder how ‘close’ to intersecting a family of size greater than 2n−1 can be. Katona, Katona and Katona introduced the idea of a ‘most probably intersecting family’. Suppose that is a family and that 0 < p < 1. Let (p) be the (random) family formed by selecting each set in independently with probability p. A family is most probably intersecting if it maximizes the probability that (p) is intersecting over all families of size ||.Katona, Katona and Katona conjectured that there is a nested sequence consisting of most probably intersecting families of every possible size. We show that this conjecture is false for every value of p provided that n is sufficiently large.