Supersaturation in Posets and Applications Involving the Container Method

TL;DR

This paper develops a framework for supersaturation problems in posets, providing bounds on comparable pairs, and applies it to classical posets to derive bounds on antichains and their properties.

Contribution

It introduces a new framework for supersaturation in posets and generalizes container methods to derive bounds on antichains and comparable pairs.

Findings

Bounds on the minimum number of comparable pairs in large subsets of posets.

Asymptotic bounds on the size of the largest antichain in random subsets.

Application of the framework to classical posets like the boolean lattice and divisor posets.

Abstract

We consider 'supersaturation' problems in partially ordered sets (posets) of the following form. Given a finite poset $P$ and an integer $m$ greater than the cardinality of the largest antichain in $P$, what is the minimum number of comparable pairs in a subset of $P$ of cardinality $m$? We provide a framework for obtaining lower bounds on this quantity based on counting comparable pairs relative to a random chain and apply this framework to obtain supersaturation results for three classical posets: the boolean lattice, the collection of subspaces of $\mathbb{F}_q^n$ ordered by set inclusion and the set of divisors of the square of a square-free integer under the 'divides' relation. The bound that we obtain for the boolean lattice can be viewed as an approximate version of a known theorem of Kleitman. In addition, we apply our supersaturation results to obtain (a) upper bounds on the number of antichains in these posets and (b) asymptotic bounds on the cardinality of the largest antichain in $p$-random subsets of these posets which hold with high probability (for $p$ in a certain range). The proofs of these results rely on a 'container-type' lemma for posets which generalises a result of Balogh, Mycroft and Treglown. We also state a number of open problems regarding supersaturation in posets and counting antichains.