A Meshalkin theorem for projective geometries

Authors:
Matthias Beck;Thomas Zaslavsky
Affiliations:
Department of Mathematical Sciences, State University of New York at Binghamton, Binghamton, NY;Department of Mathematical Sciences, State University of New York at Binghamton, Binghamton, NY
Venue:
Journal of Combinatorial Theory Series A
Year:
2003

Citing 2
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Sperner theory

Sperner theory
A shorter, simpler, stronger proof of the Meshalkin-Hochberg-Hirsch bounds on componentwise antichains

Journal of Combinatorial Theory Series A

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Abstract

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 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 Erdös's generalizations of Sperner's theorem and their LYM companions, and they generalize Rota and Harper's q-analog of Erdös's generalization.