Compressions and Probably Intersecting Families

Abstract

A familyof sets is said to beintersectingifA∩B≠ ∅ for allA,B∈. It is a well-known and simple fact that an intersecting family of subsets of [n] = {1, 2, . . .,n} can contain at most 2n−1sets. Katona, Katona and Katona ask the following question. Suppose instead⊂[n] satisfies || = 2n−1+ifor some fixedi> 0. Create a new familypby choosing each member ofindependently with some fixed probabilityp. How do we chooseto maximize the probability thatpis intersecting? They conjecture that there is a nested sequence of optimal families fori= 1, 2,. . ., 2n−1. In this paper, we show that the families [n](≥r)= {A⊂ [n]: |A| ≥r} are optimal for the appropriate values ofi, thereby proving the conjecture for this sequence of values. Moreover, we show that for intermediate values ofithere exist optimal families lying between those we have found. It turns out that the optimal families we find simultaneously maximize the number of intersecting subfamilies of each possible order.Standard compression techniques appear inadequate to solve the problem as they do not preserve intersection properties of subfamilies. Instead, our main tool is a novel compression method, together with a way of ‘compressing subfamilies’, which may be of independent interest.