Random sets and intersections

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.

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