A Meshalkin theorem for projective geometries

الأصل · EN

Let M be a family of sequences (a₁,...,aₚ) where each aₖ is a flat in a projective geometry of rank n (dimension n-1) and order q, and the sum of ranks, r(a₁) +... + r(aₚ), equals the rank of the join a₁ v... v aₚ. We prove upper bounds on |M| and corresponding LYM inequalities assuming that (i) all joins are the whole geometry and for each k<p the set of all aₖ's of sequences in M contains no chain of length l, and that (ii) the joins are arbitrary and the chain condition holds for all k. These results are q-analogs of generalizations of Meshalkin's and Erdos's generalizations of Sperner's theorem and their LYM companions, and they generalize Rota and Harper's q-analog of Erdos's generalization.