Given a family of sets .Y, where the sets in 9 admit k 'degrees of freedom', we prove that not all (k+ I)-dimensional posets are containment posets of sets in 9. 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.
When the random intersection graph G n m p proposed by Karoński, Scheinerman, and Singer-Cohen [Combin Probab Comput 8 (1999), 131-159] is compared with the independent-edge G n p , the evolutions are different under some values of m and equivalent under others. In particular, when m = n α and α > 6, the total variation distance between the graph random variables has limit 0.
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