Questions on Symmetric Chains (Shahriar Shahriari, Pomona)

The set of subsets {1, 3}, {1, 3, 4}, {1, 3, 4, 6} is a symmetric chain in the partially ordered set (poset) of subsets of {1,…,6}. It is a chain, because each of the subsets is a subset of the next one. It is symmetric because the collection has as many subsets with less than 3 elements as it has subsets with more than 3 elements (3 is half of 6, the size of the original set). It is straightforward to partition the set of all subsets of {1,…,6} into symmetric chains. Such a partition is called a symmetric chain decomposition of the poset. We are interested in the following—admittedly curious sounding—question. What is the maximum integer k, such that given any collection of k disjoint symmetric chains in the poset of subsets of a finite set, we can enlarge the collection to a symmetric chain decomposition of the poset? I don’t know the answer, but in this talk, I will discuss a special case, a number of related results and questions, and provide some background on why symmetric chain decompositions are useful.