Probably Intersecting Families are Not Nested

TL;DR

This paper investigates the structure of 'most probably intersecting families' in set theory, disproving a conjecture that such families form a nested sequence for large n and all p, revealing new insights into their configuration.

Contribution

The paper disproves the conjecture that most probably intersecting families are nested for all sizes and probabilities when n is large.

Findings

Disproves the nested structure conjecture for large n

Shows the conjecture fails for all p in the range (0,1)

Provides new understanding of intersecting family configurations

Abstract

It is well known that an intersecting family of subsets of an n-element 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 X is a family and that 0<p<1. Let X(p) be the (random) family formed by selecting each set in X independently with probability p. A family X is `most probably intersecting' if it maximises the probability that X(p) is intersecting over all families of size |X|. 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.