Summary: Degrees of Freedom Versus Dimension for Containment Orders
Department of Mathematics
Tel Aviv University
Ramat Aviv 69978, Israel
Edward R. Scheinerman2
Department of Mathematical Sciences
The Johns Hopkins University
Baltimore, Maryland 21218, U.S.A.
Given a family of sets S, where the sets in S admit k `degrees of freedom', we prove
that not all (k + 1)-dimensional posets are containment posets of sets in S. Our results
depend on the following enumerative result of independent interest: Let P(n, k) denote
the number of partially ordered sets on n labeled elements of dimension k. We show that
log P(n, k) nk log n where k is fixed and n is large.
KEY WORDS: partially ordered set, containment order, degrees of freedom, partial order dimension
AMS SUBJECT CLASSIFICATION: 06A10 (primary), 14N10 (secondary)
Let S be a family of sets. We say that a partially ordered set P has an S-containment representation
provided there is a map f : P S such that x < y iff f(x) f(y). In this case we say that P is an S-order.