Probably Intersecting Families are Not Nested

Abstract

It is well known that an intersecting family of subsets of an nelement set can contain at most 2 n−1 sets. It is natural to wonder how 'close' to intersecting a family of size greater than 2 n−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 ′ = ∅. Since no intersecting family can contain both a set A and its complement A c , it is easy to see that there is no intersecting family containing more than 2 n−1 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 n−1 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

Most Probably Intersecting Hypergraphs

“…Finally, as with the results in [10], [11] and [12], the extremal families we obtain here are simultaneously optimal for the counting problems as well, and thus we use Equation (1) to resolve the probabilistic problem. It would be very interesting to develop techniques to attack the probabilistic problem directly, as one might then find a complete solution even in the regime where the optimal family depends on the underlying probability p.…”

“…This completes the proof of Theorem 1.3, showing that the initial segment of the lexicographic order is the most probably intersecting graph up to moderate densities. Note that, as in all previously obtained results in [10] and [12], these graphs actually simultaneously maximise the number of intersecting subgraphs of all sizes, and hence the most probably intersecting graphs do not depend on p. This phenomenon fails to hold for denser graphs, but we defer this discussion until Section 4.…”

“…In the same year, Russell [11] provided some evidence towards this conjecture, by proving a result similar to Theorem 1.1, showing that there is a most probably intersecting family consisting of sets that are as large as possible. However, in a later paper with Walters [12], they used the non-nestedness of the extremal graphs in Theorem 1.2 to show that the most probably intersecting families are not nested for n i=3 n i ≤ m ≤ n i=2 n i . While the above results hold for non-uniform families, much less was known in the uniform setting.…”

“…Katona, Katona and Katona showed in [10] that for m = n−1 k−1 + 1, it is optimal to add any set to a full star, and thus L n,k (m) is again optimal. By applying i, j-compressions, Russell and Walters [12] were able to show that for any m, there is a left-compressed most probably intersecting family, but were unable to show which compressed family is optimal.…”

Removal and Stability for Erdös--Ko--Rado

“…Das, Gan and Sudakov [7] studied the supersaturation problem, determining the minimum number of disjoint pairs appearing in sufficiently sparse k-uniform families. Furthermore, a probabilistic variant of this supersaturation problem was introduced by Katona, Katona and Katona [21], and further studied by Russell [25], Russell and Walters [26] and Das and Sudakov [8].…”