Improved bounds for cross-Sperner systems

TL;DR

This paper establishes improved upper and lower bounds on the size measures of cross-Sperner systems, advancing understanding of their structure and providing counterexamples to a longstanding conjecture.

Contribution

The paper introduces new bounds for cross-Sperner systems and constructs counterexamples to a 2011 conjecture, significantly advancing the theoretical understanding.

Findings

New upper and lower bounds on sum and product measures

Construction of counterexamples to a 2011 conjecture

Significant improvement over previous bounds

Abstract

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.