Let S be a finite set of cardinality k. I consider subsets of S that I order by set inclusion. For any given k, this defines the partially ordered set S_k.

To a given partially ordered set P, I associate the smallest k such that P can be embedded in S_k (every element of P is paired with an element of S_k and the order is preserved).

What is this smallest k equal to? How efficiently can it be computed?