New bounds for union-free families of sets

Abstract

Following Frankl and Füredi [1] we say a family, F, of subsets of an n-set is weakly union-free if F does not contain four distinct sets A, B, C, D with A ∪ B = C ∪ D. If in addition A ∪ B = A ∪ C implies B = C we say F is strongly union-free. Let f(n) (g(n)) be the maximum size of strongly (weakly) union-free families. In this paper we prove the following new bounds on f and g: 2[0:31349+o(1)]n ≤ f(n) ≤ 2 [0:4998+o(1)]n and g(n) ≤ 2[0:5+o(1)]n.