TY - JOUR

T1 - Partitioning the Boolean lattice into chains of large minimum size

AU - Hsu, Tim

AU - Logan, Mark J.

AU - Shahriari, Shahriar

AU - Towse, Christopher

N1 - Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

PY - 2002

Y1 - 2002

N2 - Let 2[n] denote the Boolean lattice of order n, that is, the poset of subsets of (1,…, n) ordered by inclusion. Recall that 2[n] may be partitioned into what we call the canonical symmetric chain decomposition (due to de Bruijn, Tengbergen, and Kruyswijk), or CSCD. Motivated by a question of Füredi, we show that there exists a function d(n) ∼ 1/2 n such that for any n ≥ 0, 2[n] may be partitioned into (n⌊n/2⌋) chains of size at least d(n). (For comparison, a positive answer to Füredi's question would imply that the same result holds for some d(n) ∼ π/2 n.) More precisely, we first show that for 0 ≤ j ≤ n, the union of the lowest j+1 elements from each of the chains in the CSCD of 2[n] forms a poset Tj (n) with the normalized matching property and log-concave rank numbers. We then use our results on Tj(n) to show that the nodes in the CSCD chains of size less than 2d(n) may be repartitioned into chains of large minimum size, as desired.

AB - Let 2[n] denote the Boolean lattice of order n, that is, the poset of subsets of (1,…, n) ordered by inclusion. Recall that 2[n] may be partitioned into what we call the canonical symmetric chain decomposition (due to de Bruijn, Tengbergen, and Kruyswijk), or CSCD. Motivated by a question of Füredi, we show that there exists a function d(n) ∼ 1/2 n such that for any n ≥ 0, 2[n] may be partitioned into (n⌊n/2⌋) chains of size at least d(n). (For comparison, a positive answer to Füredi's question would imply that the same result holds for some d(n) ∼ π/2 n.) More precisely, we first show that for 0 ≤ j ≤ n, the union of the lowest j+1 elements from each of the chains in the CSCD of 2[n] forms a poset Tj (n) with the normalized matching property and log-concave rank numbers. We then use our results on Tj(n) to show that the nodes in the CSCD chains of size less than 2d(n) may be repartitioned into chains of large minimum size, as desired.

KW - Boolean lattice

KW - Chain decompositions

KW - Füredi's problem

KW - Griggs' conjecture

KW - Normalized matching property

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U2 - 10.1006/jcta.2001.3197

DO - 10.1006/jcta.2001.3197

M3 - Article

AN - SCOPUS:0036166861

VL - 97

SP - 62

EP - 84

JO - Journal of Combinatorial Theory - Series A

JF - Journal of Combinatorial Theory - Series A

SN - 0097-3165

IS - 1

ER -