A Meshalkin theorem for projective geometries

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∨⋯∨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 akʹs 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 Meshalkinʹs and Erdősʹs generalizations of Spernerʹs theorem and their LYM companions, and they generalize Rota and Harperʹs q-analog of Erdősʹs generalization.