Union-free Hypergraphs and Probability Theory

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 ∪ B = C ∪ D. We prove that 2(n-log3)/3 - 2 ≤ F(n) ≤ 2(3n+2)/4. We use probability theory for the proof of both the lower and upper bounds. Some related problems are considered, too.