5 - Matchings, symmetric chain orders, and the partition lattice

Summary

This chapter should be considered as a link between the preceding flow-theoretic (resp. linear programming) approach and the succeeding algebraic approach to Sperner-type problems. The main ideas are purely combinatorial. The powerful method of decomposing a poset into symmetric chains not only provides solutions of several problems, it is also very helpful for the insight into the algebraic machinery in Chapter 6. In particular, certain strings of basis vectors of a corresponding vector space will behave like (symmetric) chains.

Definitions, main properties, and examples

Throughout let P be a ranked poset of rank n : r(P). We say that the level N_i can be matched into the level N_i+1 (resp. N_i-1) if there is a matching of size W_i in the Hasse diagram G_i = (P_i, E_i) (resp. G_i-1 = (P_i-1, E_i-1)) of the {i, i + 1}- (resp. {i – 1, i}-) rank-selected subposet (recall that a matching is a set of pairwise nonadjacent edges or arcs).

Theorem 5.1.1. If there is an index h such that N_i can be matched into N_i+1 for 0 ≤ i ≤ h and N_i can be matched into N_i–1 for h < i ≤ n, then P has the Sperner property.

Proof. If we join the arcs of the corresponding matchings at all points that are both starting point and endpoint, we obtain a partition of P into saturated chains (isolated points are considered as 1-element chains). Each such chain has a common point with N_h. Consequently, we have W_h chains and Dilworth's Theorem 4.0.1 implies d(P) ≤ W_h, that is, d(P) = W_h.