Chains, antichains, and fibres

Abstract

Call a subset of an ordered set a fibre if it meets every maximal antichain. We prove several instances of the conjecture that, in an ordered set P without splitting elements, there is a subset F such that both F and P − F are fibres. For example, this holds in every ordered set without splitting points and in which each chain has at most four elements. As it turns out several of our results can be cast more generally in the language of graphs from which we may derive “complementary” results about cutsets of ordered sets, that is, subsets which meet every maximal chain. One example is this: In a finite graph G every minimal transversal is independent if and only if G contains no path of length three.