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A Kruskal–Katona Type Theorem for the Linear Lattice

✍ Scribed by S Bezrukov; A Blokhuis

Publisher
Elsevier Science
Year
1999
Tongue
English
Weight
154 KB
Volume
20
Category
Article
ISSN
0195-6698

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✦ Synopsis

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 find 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 first 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 case = 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 case = 2 and k = 1.

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