A resolution of the Aharoni-Korman conjecture

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 $\bigoplus_{x\inω} Q_x$, 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.