Probably Intersecting Families are Not Nested

It is well known that an intersecting family of subsets of an nelement 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 A is a family and that 0 < p < 1. Let A(p) be the (random) family formed by selecting each set in A independently with probability p. A family A is most probably intersecting if it maximises the probability that A(p) is intersecting over all families of size |A|. 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. We start by recalling the definition of an intersecting family : we say that A ⊂ P([n]) is intersecting if for any A,A ∈ A we have A∩A′ 6= ∅. Since no intersecting family can contain both a set A and its complement A, it is easy to see that there is no intersecting family containing more than 2 sets. We remark that this upper bound is tight and that, in fact, any intersecting family can be extended to an intersecting family of this size. Having observed this bound, it is natural to wonder how ‘close’ to intersecting a family of size greater than 2 can be. Katona, Katona and Katona [4] introduced the idea of a most probably intersecting family. Suppose that A is a family and that 0 < p < 1. Let A(p) be the (random) family Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, England. [email protected] School of Mathematical Sciences, Queen Mary, University of London, London E1 4NS, England [email protected]