Union-Free Families of Sets and Equations over Fields

Abstract Let X be an n-element set and F ⊂ (kx) such that all the (2 | F |) sets F 1 ⌣ F 2 , F1, F2 ∈ F are distinct. Solving a problem of P. Erdős (“Proceedings, 8th Southeastern Conf. on Combinatorics, Graph Theory, and Computing, Baton Rouge, 1977”, pp.3–12) we show that there exist positive constants ck, c′k such that ckn⌈4k/3⌉/2⩽| F |⩽c′kn⌈4k/3⌉/2 holds. For the proof of the lower bound we need a theorem of independent interest which is of algebraic number-theoretic character (Theorem 1.4.).