Given a family of sets L, where the sets in L admit k 'degrees of freedom', we prove that not all (k+1)-dimensional posets are containment posets of sets in L. 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.
- AMS subject classifications (1980): 06A10 (primary), 14N10 (secondary)
- Partially ordered set
- containment order
- degrees of freedom
- partial order dimension