Compressions and Probably Intersecting Families

TL;DR

This paper investigates the structure of large intersecting families of sets, proving a conjecture about optimal families for maximizing intersection probability after random sampling, using novel compression techniques.

Contribution

It introduces a new compression method to identify optimal families of sets, confirming a conjecture for certain sizes and exploring intermediate cases.

Findings

Families of the form [n]^( extgreater r) are optimal for specific sizes.

Optimal families maximize the number of intersecting subfamilies of all orders.

Standard compression techniques are inadequate; a new method is developed.

Abstract

A family X of sets is said to be intersecting if any two members of X have non-empty intersection. It is a well-known and simple fact that an intersecting family of subsets of [n]={1,2,...,n} can contain at most 2^(n-1) sets. Katona, Katona and Katona ask the following question. Suppose instead a family X of subsets of [n] satisfies |X|=2^(n-1)+i for some fixed i>0. Create a new family X_p by choosing each member of X independently with some fixed probability p. How do we choose X to maximize the probability that X_p is intersecting? They conjecture that there is a nested sequence of optimal families for i=1, 2, ..., 2^(n-1). In this paper, we show that the families [n]^(\ge r)={A\subset[n]:|A|\ge r} are optimal for the appropriate values of i, thereby proving the conjecture for this sequence of values. Moreover, we show that for intermediate values of i there 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 every 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.