Improved bounds for cross-Sperner systems

A collection of families \((\mathcal{F}_{1}, \mathcal{F}_{2} , \cdots , \mathcal{F}_{k}) \in \mathcal{P}([n])^k\) is cross-Sperner if there is no pair \(i \not= j\) for which some \(F_i \in \mathcal{F}_i\) is comparable to some \(F_j \in \mathcal{F}_j\). Two natural measures of the `size' of such a family are the sum \(\sum_{i = 1}^k |\mathcal{F}_i|\) and the product \(\prod_{i = 1}^k |\mathcal{F}_i|\). We prove new upper and lower bounds on both of these measures for general \(n\) and \(k \ge 2\) which improve considerably on the previous best bounds. In particular, we construct a rich family of counterexamples to a conjecture of Gerbner, Lemons, Palmer, Patk\'{o}s, and Sz\'{e}csi from 2011.