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Maximal antichains of minimum size

Kalinowski, Thomas, Leck, Uwe and Roberts, Ian T. (2013). Maximal antichains of minimum size. Electronic Journal of Combinatorics,20(2):P3-1-P3-14.

Document type: Journal Article
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IRMA ID 82794376xPUB254
Title Maximal antichains of minimum size
Author Kalinowski, Thomas
Leck, Uwe
Roberts, Ian T.
Journal Name Electronic Journal of Combinatorics
Publication Date 2013
Volume Number 20
Issue Number 2
ISSN 1077-8926   (check CDU catalogue open catalogue search in new window)
Scopus ID 2-s2.0-84876216460
Start Page P3-1
End Page P3-14
Total Pages 14
Place of Publication United States of America
Publisher Electronic Journal of Combinatorics
HERDC Category C1 - Journal Article (DIISR)
Abstract Let n⩾4 be a natural number, and let K be a set K⊆[n]:={1,2,…,n} . We study the problem of finding the smallest possible size of a maximal family A of subsets of [n] such that A contains only sets whose size is in K , and A⊈B for all {A,B}⊆A , i.e. A is an antichain. We present a general construction of such antichains for sets K containing 2, but not 1. If 3∈K our construction asymptotically yields the smallest possible size of such a family, up to an o(n 2 ) error. We conjecture our construction to be asymptotically optimal also for 3∉K , and we prove a weaker bound for the case K={2,4} . Our asymptotic results are straightforward applications of the graph removal lemma to an equivalent reformulation of the problem in extremal graph theory, which is interesting in its own right.

Keywords Extremal set theory
Sperner property
Maximal antichains
Flat antichains
Open access True
Additional Notes This is an Open Access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Description for Link Link to published version

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