Mathematics Colloquium
Spring 2010
William T. Trotter
Georgia Institute of Technology
Chains and Antichains in Finite Partially Ordered Sets
There are several instances of dual (or nearly dual) theorems involving chains and antichains in finite partially ordered sets. We analyze four interesting cases: (1) Dilworth's theorem and the Greene/Kleitman generalizations; (2) On-line chain and antichain partitioning; (3) families of disjoint maximal chains/antichains; and (4) fibers and co-fibers. The first topic is classic and certainly belongs to the core of combinatorial mathematics, while there are quite new and appealing results on the second and third topics. For the last, there are difficult open problems, but the recent work on the other areas may shed new light on how they should be attacked.