Fishbone conjecture
If [math](P,\lt)[/math] is a partial order without infinite antichains, then there is a partition [math]\mathcal{A}[/math] of [math] P [/math] into antichains in such a way that there is a chain [math]C[/math] that intersects every element of [math]\mathcal{A}[/math].
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 [math]2[/math]. Excluding infinite antichains is necessary, as the following example shows: there is a chain [math]C_n[/math] of length [math]n[/math] for every [math]n[/math], these chains are pairwise disjoint, and elements from distinct [math]C_n[/math] are incomparable. The following dual statement is known ([1] p. 20. Theorem 7.1).
Theorem If [math](P,\lt)[/math] is a partial order without infinite chains, then there is a partition [math]\mathcal{C}[/math] of [math] P [/math] into chains in such a way that there is a antichain [math]A[/math] that intersects every element of [math]\mathcal{C}[/math].
The proof is based on the infinite version of Kőnig's theorem and the usual bipartite graph representation of [math](P,\lt)[/math].
References
- ↑ Diestel, Reinhard, C. Nash-Williams, and J. A. St. Directions in infinite graph theory and combinatorics, North-Holland, 1992. author link
