Sperner systems consisting of pairs of complementary subsets

Abstract

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 $\text{A}$ = {;A₁,…, A_l}; consists of k-subsets of a set with n > 2k elements such that A_i ∩ A_j ≠ ∅ for all i, j then l ⩽ (_k−1ⁿ⁻¹). Schönheim proved that if A₁, …, A_l are subsets of a set S with n elements such that A_i ∉ A_j, A_i ∩ A_j ≠ ø and A_i ∪ A_j ≠ S for all i ≠ j then $\text{l ⩽ (}{}_{\mathrm{<mtext>[</mtext><mtext>n</mtext><mtext>2</mtext><mtext>]\; -\; 1</mtext>}}{}^{\mathrm{n\; -\; 1}}\text{)}$. In this note we prove a common strengthening of these results.