Sizes of Flat Maximal Antichains of Subsets

Jerrold R. Griggs; Thomas Kalinowski; Uwe Leck; Ian T. Roberts; Michael Schmitz

doi:10.1007/s11083-024-09675-9

Sizes of Flat Maximal Antichains of Subsets

Jerrold R. Griggs, Thomas Kalinowski, Uwe Leck, Ian T. Roberts, Michael Schmitz

Faculty of Arts and Society

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Abstract

This is the second of two papers investigating for which positive integers m there exists a maximal antichain of size m in the Boolean lattice B_n (the power set of [n]:={1,2,⋯,n}, ordered by inclusion). In the first part, the sizes of maximal antichains have been characterized. Here we provide an alternative construction with the benefit of showing that almost all sizes of maximal antichains can be obtained using antichains containing only l-sets and (l+1)-sets for some l.

Original language	English
Pages (from-to)	1-26
Number of pages	26
Journal	Order
DOIs	https://doi.org/10.1007/s11083-024-09675-9
Publication status	E-pub ahead of print - 27 Jun 2024

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title = "Sizes of Flat Maximal Antichains of Subsets",

abstract = "This is the second of two papers investigating for which positive integers m there exists a maximal antichain of size m in the Boolean lattice Bn (the power set of [n]:={1,2,⋯,n}, ordered by inclusion). In the first part, the sizes of maximal antichains have been characterized. Here we provide an alternative construction with the benefit of showing that almost all sizes of maximal antichains can be obtained using antichains containing only l-sets and (l+1)-sets for some l.",

keywords = "Antichain, Flat Antichain, Kruskal-Katona theorem, Saturation spectrum, Sperner family",

author = "Griggs, {Jerrold R.} and Thomas Kalinowski and Uwe Leck and Roberts, {Ian T.} and Michael Schmitz",

year = "2024",

month = jun,

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doi = "10.1007/s11083-024-09675-9",

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T1 - Sizes of Flat Maximal Antichains of Subsets

AU - Griggs, Jerrold R.

AU - Kalinowski, Thomas

AU - Leck, Uwe

AU - Roberts, Ian T.

AU - Schmitz, Michael

PY - 2024/6/27

Y1 - 2024/6/27

N2 - This is the second of two papers investigating for which positive integers m there exists a maximal antichain of size m in the Boolean lattice Bn (the power set of [n]:={1,2,⋯,n}, ordered by inclusion). In the first part, the sizes of maximal antichains have been characterized. Here we provide an alternative construction with the benefit of showing that almost all sizes of maximal antichains can be obtained using antichains containing only l-sets and (l+1)-sets for some l.

AB - This is the second of two papers investigating for which positive integers m there exists a maximal antichain of size m in the Boolean lattice Bn (the power set of [n]:={1,2,⋯,n}, ordered by inclusion). In the first part, the sizes of maximal antichains have been characterized. Here we provide an alternative construction with the benefit of showing that almost all sizes of maximal antichains can be obtained using antichains containing only l-sets and (l+1)-sets for some l.

KW - Antichain

KW - Flat Antichain

KW - Kruskal-Katona theorem

KW - Saturation spectrum

KW - Sperner family

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SN - 0167-8094

SP - 1

EP - 26

JO - Order

JF - Order

ER -

Sizes of Flat Maximal Antichains of Subsets

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