Minimizing the Number of Unions

Abstract

For a given number of k-sets, how should we choose them so as to minimize the union-closed family that they generate? Our main aim in this paper is to show that, if A is a family of k-sets of size(Formula Presented), and t is sufficiently large, then the union-closed family generated by A has size at least that generated by the family of all k-sets from a t-set. This proves (for this size of family) a conjecture of Roberts. We also make some related conjectures, and give some other results, including a new proof of the result of Leck, Roberts and Simpson that exactly determines this minimum (for all sizes of the family) when k = 2, as well as resolving the conjecture of Roberts when the size of the family is very close to (Formula Presented).