On the decomposition of order-separable posets of countable width into chains (Q1568672)

A poset \(X\) is order-separable if it contains a countable dense subset; \(X\) has countable width if it contains no uncountable antichain. The following question is posed: Can every poset with these two properties be written as the union of a countable number of chains? The authors show that the answer is ``no'' if there is a 2-entangled subset of \(\mathbb{R}\), i.e. an uncountable subset \(A\subset \mathbb{R}\) such that there is no uncountable monotone function from some subset of \(A\) into \(A\) with no fixed points, and it is ``yes'' under the Open Coloring Axiom.