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 ℵ is a family and that 0<p<1. Let ℵp be the (random) family formed by selecting each set in ℵ independently with probability p. A family ℵ is most probably intersecting if it maximises the probability that ℵp is intersecting over all families of size |ℵ|. 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.
Paper (pdf)