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- W1654314556 abstract "AbstractLet n > 3 be a natural number. We study the problem to nd the smallest r suchthat there is a family Aof 2-subsets and 3-subsets of [n] = f1;2;:::;ngwith the followingproperties: (1) Ais an antichain, i.e. no member of Ais a subset of any other member of A,(2) Ais maximal, i.e. for every X 22 [n] nAthere is an A 2Awith X A or A X, and(3) Ais r-regular, i.e. every point x 2[n] is contained in exactly r members of A. We provelower bounds on r, and we describe constructions for regular maximal antichains with smallregularity. 1 Introduction Let n> 3 be an integer and [n] := f1;2;:::;ng. By 2 [n] we denote the family of all subsets of [n]and by [n]k the family of all k-subsets of [n]. A family A2 [n] is an antichain if A6Bfor alldistinct A;B2A. An antichain Ais called at if it is contained in two consecutive levels of 2 [n] ,i.e. if A [n]k [ [n]k+1 for some k. The remarkable Flat Antichain Theorem, which follows fromresults of Lieby [13] (see also [14]) and Kisvolcsey [12], says that for every antichain Athere is anequivalent at antichain A" @default.
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- W1654314556 date "2012-06-17" @default.
- W1654314556 modified "2023-09-27" @default.
- W1654314556 title "Minimizing the regularity of maximal regular antichains of 2- and 3-sets" @default.
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