DOIAbstract
AbstractWe present an analog of the well-known Kruskal–Katona theorem for the poset of subspaces of PG(n,2) ordered by inclusion. For givenk,ℓ (k<ℓ) andmthe problem is to find a family of sizemin the set of ℓ-subspaces of PG(n,2), containing the minimal number ofk-subspaces. We introduce two lexicographic type orders O1and O2on the set of ℓ-subspaces, and prove that the firstmof them, taken in the order O1, provide a solution in the casek=0 and arbitrary ℓ>0, and one taken in the order O2, provide a solution in the case ℓ=n−1 and arbitraryk<n−1. Concerning other values ofkand ℓ, we show that forn≥ 3 the considered poset is not Macaulay by constructing a counterexample in the case ℓ=2 andk=1
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