The Aharoni--Korman conjecture for $N$-free posets with no infinite antichain

We give a necessary and sufficient condition for a $P_4$-free graph to be a cograph. This allows us to obtain a simple proof of the fact that finite $P_4$-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 every $N$-free poset with no infinite antichain has a chain and a partition into antichains so that each part intersects the chain. This answers a conjecture of Aharoni and Korman (Order $\mathbf{\text{9}}$ (1992) 245–253) in this case.