Thicknesses of ordered sets.

The thickness of a finite poset (partially ordered set) is the cardinality of the smallest portion of the poset into induced subposets of dimension two or less. Thickness combines aspects of width and length. It never exceeds length, can equal length for any finite length, and never exceeds about half the width. There are dimension 3 posets with arbitrarily large thicknesses, and there are thickness 2 posets with arbitrarily large dimension. For m>= 3, the m-dimensional poset of the singletons and their complements of an m-set ordered by inclusion has thickness 2; the thickness of the m-dimensional poset of all subsets of an m-set ordered by inclusion goes to infinity as m gets large.