A Meshalkin theorem for projective geometries

LetM 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 ∨ · · · ∨ 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 Erdős’s generalizations of Sperner’s theorem and their LYM companions, and they generalize Rota and Harper’s q-analog of Erdős’s generalization.