Some progress on the Aharoni–Korman conjecture

Abstract

Aharoni and Korman (1992) [3] conjectured that every ordered set with no infinite antichains possesses a chain and a partition into antichains so that each part intersects the chain.The conjecture is verified for posets whose incomparability graph is locally finite. It follows that the conjecture is true for (3+1)-free posets with no infinite antichains.We give a necessary and sufficient condition for a P4-free graph to be a cograph. This allows us to obtain a simple proof of the fact that finite P4-free graphs are finite cographs. We also prove that N-free chain complete posets and N-free posets with no infinite antichains are series-parallel.As a consequence, we obtain that the conjecture is true for N-free posets with no infinite antichain.