Europ. J. Combinatorics (1984) 5, 127-131 Union-free Hypergraphs and Probability Theory PETER FRANKL AND ZOLTAN FUREDI Let F(n) denote the maximum number of distinct subsets of an n-element set such that there are no four distinct subsets: A, B, C, D with A v B = C v D. We prove that 2<n-Ios 3 ll 3 - 2.;:; F( n).;:; 2< 3 n+Z)/ 4 • We use probability theory for the proof of both the lower and upper bounds. Some related problems are considered, too. 1. INTRODUCTION In 1969 Erdos and Moser [4] raised the problem of estimating f(n), the maximum number of distinct subsets of an n-element set such that all the f"<;>) pairwise unions are different. THEOREM 1. 1 + 2 (n+l)/2. (1) Notice that the upper bound is an immediate consequence of (I<;J) :s;; 2n. To prove the lower bound we use an algebraic construction which is a modification of a construction of Babai and S6s [1]. How a family of sets can fail to have the union-free property? There are essentially two possibilities: (a) there are four distinct sets A, B, C, D with Au B =CuD. (b) there are three distinct sets A, B, C with Au B = Au C. We call families for which (a) never holds weakly union-free, and those for which (b) never holds cancellative (the second name indicates that Au B =Au C implies B = C). We denote by F(n)(G(n)) the maximum number of subsets of an n-set in a weakly union-free (cancellative) family, respectively. Our main result is the following: THEOREM 2. (2) The lower bound is deduced by a non-constructive, probabilistic method. The proof of the upper bound uses information theory, it was inspired by the paper Kleitman, Shearer and Sturtevant [9]. For cancellative families we prove: THEOREM 3. (8/9)'<n> 13 3n 13 G(n) < n IS (n :2l: 14), (3) where e( n) is determined by 0 e( n) 2, n + e( n) is divisible by 3. Erdos and Katona (cf. [8]) conjecture that the lower bound is exact. Their construction is simple: let ... , Xq be pairwise disjoint sets with union of size n with !Xi!= 2 or 3 and with at most two sets of size 2 among the Xi. Let our family consist of all the transversals that is of those sets which intersect each Xi in one element. Clearly this family achieves the lower bound and it is cancellative. 2. RELATED AND OPEN PROBLEMS Let k be an integer, k:2l:2. Let us denote by fk(n) the maximum number of k-subsets of ann-set forming a union-free family, Fk(n), Gk(n) are defined similarly. Then j 2 (n), 127 0195-6698/84/020127 +05 $02.00/0 © 1984 Academic Press Inc. (London) Limited