A resolution of the Aharoni-Korman conjecture

TL;DR

This paper disprove the Aharoni-Korman conjecture in general but confirms it for specific classes of countable posets, advancing understanding of the structure of FAC posets and their chains.

Contribution

It provides a counterexample to the conjecture and proves its validity for a broad class of countable posets with specific interval avoidance properties.

Findings

Counterexample to the conjecture in full generality.

Proof of the conjecture for certain countable posets.

Investigation of strongly maximal chains in FAC posets.

Abstract

A poset $P$ is said to satisfy the finite antichain condition, or FAC for short, if it has no infinite antichain. It was conjectured by Aharoni and Korman in 1992 that any FAC poset $P$ possesses a chain $C$ and a partition into antichains such that $C$ meets every antichain of the partition. Our main results are twofold. We provide a counterexample to the conjecture in full generality, but, despite this, we also prove that the conjecture does hold true for a broad class of posets. In particular, we prove that the Aharoni-Korman conjecture holds for countable posets avoiding intervals $I$ such that either $I$ or its reverse $I^*$ is of the form $\bigoplus_{x\in\omega} Q_x$, where each $Q_x$ is infinite and co-wellfounded. In pursuit of these goals, we also investigate other facets of the structure of FAC posets. In particular, we consider strongly maximal chains in FAC posets, proving some results, and posing several questions and conjectures.