In what ways can we partition a partially ordered set (poset) into linearly ordered subsets (chains)? We will report on some recent progress on a number of old conjectures.
In particular, two chains C1 and C2 in a finite ranked poset P (a finite poset is ranked if all maximal chains have the same size) are said to be nested if |C1| ≤ |C2| implies that the levels occurring in C1 are a subset of the levels occurring in C2.
A thirty-year old conjecture of Griggs gives a sufficient condition---the so-called normalized matching condition, also known as the LYM property---for guaranteeing a decomposition of a poset into pairwise nested chains.
In this talk, we will survey what is known---including some new results---on partitioning normalized matching posets into chains.