Fishbone conjecture

If $(P,\lt)$ is a partial order without infinite antichains, then there is a partition $\mathcal{A}$ of $P$ into antichains in such a way that there is a chain $C$ that intersects every element of $\mathcal{A}$.

Remarks

This was conjectured by R. Aharoni and V. Korman and they have proved it in the special case when every antichain has size at most $2$. Excluding infinite antichains is necessary, as the following example shows: there is a chain $C_n$ of length $n$ for every $n$, these chains are pairwise disjoint, and elements from distinct $C_n$ are incomparable. The following dual statement is known ([1] p. 20. Theorem 7.1).

Theorem If $(P,\lt)$ is a partial order without infinite chains, then there is a partition $\mathcal{C}$ of $P$ into chains in such a way that there is a antichain $A$ that intersects every element of $\mathcal{C}$.

The proof is based on the infinite version of Kőnig's theorem and the usual bipartite graph representation of $(P,\lt)$.

References

1. Diestel, Reinhard, C. Nash-Williams, and J. A. St. Directions in infinite graph theory and combinatorics, North-Holland, 1992. author link