Improved bounds for cross-Sperner systems

Abstract

A collection of families (Fthere is no pair i = j for which some F i ∈ F i is comparable to some F j ∈ F j . Two natural measures of the 'size' of such a family are the sum k i=1 |F i | and the product k i=1 |F i |. We prove new upper and lower bounds on both of these measures for general n and k ≥ 2 which improve considerably on the previous best bounds. In particular, we construct a rich family of counterexamples to a conjecture of Gerbner, Lemons, Palmer, Patkós, and Szécsi from 2011.