The Aharoni--Korman conjecture, also known as the fishbone conjecture, states that any poset contains a chain C and a partition into antichains such that C meets every antichain in the partition. Our results are twofold. Firstly, we construct a poset for which the conjecture is false. Secondly, we demonstrate that this counterexample is, in some sense, minimal, giving a strong positive result which shows that the conjecture is true if one makes an additional assumption about the structure of the poset.
