@Book{1987259742,
author="Gr{\"u}ttm{\"u}ller, Martin
and Kalinowski, Thomas
and Leck, Uwe
and Hartmann, Sven
and Roberts, Ian T.",
title="Maximal flat antichains of minimum weight",
series="Preprints aus dem Institut f{\"u}r Mathematik",
year="2005",
address="Germany; Rostock",
abstract="We study maximal families A of subsets of [n] = {\{}1; 2; : : : ; n{\}} such that A contains only pairs and triples and A /⊆ B for all {\{}A,B{\}} ⊆ A, i.e. A is an antichain. For any n, all such families A of minimum size are determined. This is equivalent to finding all graphs G = (V,E) with |V| = n and with the property that every edge is contained in some triangle and such that |E| - |T| is maximum, where T denotes the set of triangles in G. The largest possible value of |E| - |T| turns out to be equal to ⌊(n+1)^2/8⌋. Furthermore, if all pairs and triples have weights w2 and w3, respectively, the problem of minimizing the total weight w(A) of A is considered. We show that min w(A) = (2w2 + w3)n^2/8 + o(n^2) for 0 < w3/w2 =: $\lambda$ = $\lambda$(n) < 2. For $\lambda$ ≥ 2 our problem is equivalent to the (6,3)-problem of Ruzsa and Szemer{\'e}di, and by a result of theirs it follows that min w(A) = w2n^2/2 + o(n^2).<eng>",
note="Martin Gr{\"u}ttm{\"u}ller, Thomas Kalinowski, Uwe Leck, Sven Hartmann, Ian T. Roberts",
doi="10.18453/rosdok_id00005898",
url="https://purl.uni-rostock.de/rosdok/id00005898",
url="https://doi.org/10.18453/rosdok_id00005898",
language="English"
}