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].
A countable conterexample for the conjecture has been found by Lawrence Hollom [1]. He also showed that the conjecture is true for countable posets whose contiguous chains satisfy an additional condition.
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 ([2] 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
- ↑ Hollom, Lawrence, A resolution of the Aharoni-Korman conjecture, arXiv:2411.16844 (2024), arXiv link
- ↑ Diestel, Reinhard, C. Nash-Williams, and J. A. St. Directions in infinite graph theory and combinatorics, North-Holland, 1992. author link
