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Matthias Beck

A short, generalizable, and q-analogizable proof of Meshalkin's generalization of Sperner's theorem on componentwise antichains

Abstract for the Combinatorics and Number Theory Seminar 2001 September 12

We study classes of ordered partitions into p parts of an n-element set, in which for each k the family of k-th parts is an antichain (that is, a family of sets in which there are no subset relations). Meshalkin's theorem, a generalization of Sperner's Theorem, states that the maximum size of such a class is the maximum size of a multinomial coefficient \binom{n}{a_1,…,a_p}. We generalize this and a stronger inequality called an LYM inequality. Our proof is simpler as well as more general than all previous proofs and extends to analogous statements–with a twist–about flats in a projective geometry.

seminars/comb/abstract.200109beck.txt · Last modified: 2020/01/29 14:03 (external edit)