Probably intersecting families are not nested

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 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.