Sperner systems consisting of pairs of complementary subsets


It was proved by Erdös, Ko, and Radó (Intersection theorems for systems of finite sets, Quart. J. Math. Oxford Ser. 12 (1961), 313-320.) that if A = {;A1,..., Al}; consists of k-subsets of a set with n > 2k elements such that Ai ∩ Aj ≠ ∅ for all i, j then l ≤ (k-1n-1). Schönheim proved that if A1, ..., Al are subsets of a set S with n elements such that Ai ∉ Aj, Ai ∩ Aj ≠ ø and Ai ∪ Aj ≠ S for all i ≠ j then l ≤ ([ n 2] - 1n - 1). In this note we prove a common strengthening of these results. © 1973.

Publication Title

Journal of Combinatorial Theory, Series A