The Width of Downsets

TL;DR

This paper investigates the maximum size of antichains within downsets of the subset lattice, providing new decomposition theorems and bounds, extending classical Sperner's theorem to more general structures.

Contribution

It introduces a Dilworth-type decomposition theorem for downsets and offers a new proof for the maximum antichain size in arbitrary downsets, advancing understanding beyond Sperner's theorem.

Findings

Established a Dilworth-type decomposition theorem for downsets

Derived bounds for maximum antichain sizes in downsets

Proved a conjecture on minimal antichain sizes in certain downsets

Abstract

How large an antichain can we find inside a given downset in the lattice of subsets of [n]? Sperner's theorem asserts that the largest antichain in the whole lattice has size the binomial coefficient C(n, n/2); what happens for general downsets? Our main results are a Dilworth-type decomposition theorem for downsets, and a new proof of a result of Engel and Leck that determines the largest possible antichain size over all downsets of a given size. We also prove some related results, such as determining the maximum size of an antichain inside the downset that we conjecture minimizes this quantity among downsets of a given size.