An algorithmic construction of union-intersection-bounded families

Abstract

In this paper, we present lower bounds and algorithmic constructions of union-intersection-bounded families of sets. The lower bound is established using the Lovász Local Lemma. This bound matches the best known bound for the size of union-intersection-bounded families of sets. We then use the variable framework for the Lovász Local Lemma, to discuss an algorithm that outputs explicit constructions that attain the lower bound. The algorithm has polynomial complexity in the number of points in the family.