A Kruskal - Katona Type Theorem for the
نویسنده
چکیده
We present an analog of the well-known Kruskal-Katona theorem for the poset of subspaces of PG (n; 2) ordered by inclusion. For given k; ` (k < `) and m the problem is to nd a family of size m in the set of`-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`-subspaces, and prove that the rst m of them, taken in the order O 1 , provide a solution in the case k = 0 and arbitrary`> 0, and one taken in the order O 2 , provide a solution in the casè = n ? 1 and arbitrary k < n ? 1. Concerning other values of k and`, we show that for n 3 the considered poset is not Macaulay by constructing a counterexample in the casè = 2 and k = 1.
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