Partitioning the boolean lattice into a minimal number of chains of relatively uniform size

Authors:
Tim Hsu;Mark J. Logan;Shahriar Shahriari;Christopher Towse
Affiliations:
Department of Mathematics, San José State University, San José, CA;Department of Mathematics, University of Minnesota, Morris, MN;Department of Mathematics, Pomona College, Claremont, CA;Department of Mathematics, Scripps College, Claremont, CA
Venue:
European Journal of Combinatorics
Year:
2003

Citing 4
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Sperner theory

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Partitioning the Boolean lattice into chains of large minimum size

Journal of Combinatorial Theory Series A

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Abstract

Let 2[n] denote the Boolean lattice of order n, that is, the poset of subsets of {1,..., n} ordered by inclusion. Extending our previous work on a question of Füredi, we show that for any c 1, there exist functions e(n) ∼ √n/2 and f(n)∼ c√n log n and an integer N (depending only on c) such that for all n , there is a chain decomposition of the Boolean lattice 2[n] into (n ⌊n/2⌋) chains, all of which have size between e(n) and f(n). (A positive answer to Füredi's question would imply that the same result holds for some e(n) ∼ √π/2 √n and f(n) = e(n) + 1.) The main tool used is an apparently new observation about rank-collection in normalized matching (LYM) posets.