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

  • Authors:

  • Matthias Beck;Thomas Zaslavsky

  • Affiliations:

  • Department of Mathematical Sciences, State University of New York at Binghamton, Binghamton, New York;Department of Mathematical Sciences, State University of New York at Binghamton, Binghamton, New York

  • Venue:
  • Journal of Combinatorial Theory Series A
  • Year:
  • 2002

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Abstract

Meshalkin's theorem states that a class of ordered p-partitions of an n-set has at most max(n a1.......ap) members if for each k the kth parts form an antichain. We give a new proof of this and the corresponding LYM inequality due to Hochberg and Hirsch, which is simpler and more general than previous proofs. It extends to a common generalization of Meshalkin's theorem and Erdös's theorem about r-chain-free set families.