The chain covering number of a poset with no infinite antichains

TL;DR

This paper characterizes the minimal chain covering numbers of posets with no infinite antichains, providing explicit lists of posets that embed into any such poset with a given covering number.

Contribution

It offers a complete classification of posets with no infinite antichains based on their chain covering number, extending previous results to all infinite successor cardinals and certain limit cardinals.

Findings

For successor cardinals, the list has two elements: $[ u]^2$ and its dual.

For limit cardinals with weakly compact cofinality, the list has four elements.

The classification applies to all posets with no infinite antichains, linking embedding properties to chain covering numbers.

Abstract

The chain covering number $\Cov(P)$ of a poset $P$ is the least number of chains needed to cover $P$. For a cardinal $\nu$, we give a list of posets of cardinality and covering number $\nu$ such that for every poset $P$ with no infinite antichain, $\Cov(P)\geq \nu$ if and only if $P$ embeds a member of the list. This list has two elements if $\nu$ is a successor cardinal, namely $[\nu]^2$ and its dual, and four elements if $\nu$ is a limit cardinal with $\cf(\nu)$ weakly compact. For $\nu= \aleph_1$, a list was given by the first author; his construction was extended by F. Dorais to every infinite successor cardinal $\nu$.