Width and f-vectors of Cutsets in the Truncated Boolean Lattice

In this paper we will survey a collection of recent results about chains and cutsets in the Boolean lattice and some other posets. In particular, we consider the possibilities for the width (i.e., the size of the largest anti-chain) and the f-vectors (i.e., the number of subsets of various sizes) of cutsets. Given k chains in the Boolean lattice, we will also consider the possible number of pair-wise disjoint maximal chains that do not intersect the given k chains.