We present an analog of the well-known Kruskal-Katona theorem For the poser of subspaces of PG(n, 2) ordered by inclusion. For given k, l (k < l) and in the problem is to find a family of size in in the set of l-subspaces of PG(n, 2), containing the minimal number of k-subspaces. We introduce two lexicographic type orders O-1 and O-2 on the set of l-subspaces, and prove that the first m of them, taken in the order O-1, provide a solution in the case k = 0 and arbitrary l > 0, and one taken in the order O-2, provide a solution in the case l = n - 1 and arbitrary k < n - 1. Concerning other values of k and l, we show that for n greater than or equal to 3 the considered poser is not Macaulay by constructing a counterexample in the case l = 2 and k = 1. (C) 1999 Academic Press.
