Random sets and intersections

TL;DR

This paper derives a simple formula for the probability that a subset intersects each part of a partition in at least a certain number of elements, connecting combinatorial set intersection problems with classical balls-and-bins models.

Contribution

It introduces an elementary approach to estimate intersection probabilities in partitions, linking these to well-known balls-and-bins asymptotics under broad conditions.

Findings

Derived an easy-to-use formula for intersection probabilities

Established asymptotic equivalence with balls-and-bins minimum problems

Applied results to problems related to Pascal's triangle adic transformation

Abstract

The following class of problems arose out of vain attempts to show that the Pascal's triangle adic transformation has trivial spectrum. Partition a set of size $N$ into sets of size $S \equiv S(N)$ (ignoring leftovers). What is the likelihood that a set of size $K \equiv K(N)$ will intersect each set in the partition in at least $R \equiv R(N)$ members (as $N$ increases)? Via elementary techniques and under reasonable hypotheses, we obtain an easy-to-use formula. Although different from the corresponding minimum problem for balls and bins (with $m = K$ balls and $n = N/S$ bins), under modest constraints, the asymptotic probabilities are the same.