The structure of FAC posets and 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. Such posets exhibit rich and complex structure, and 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. While this conjecture is now known to be false, in this paper we 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 containing no saturated chain $D$ such that either $D$ or its reverse $D^{\ast }$ is of the form ${⨁}_{x\in \omega }{D}_x$, where each $D_x$ is infinite and co-wellfounded. In pursuit of this goal, we prove several structural results, the foremost of which demonstrates how a countable FAC poset may be broken up into a collection of scattered posets which reflect the structure of the poset as a whole.