10.18453/rosdok_id00005898 Grüttmüller, Martin Martin Grüttmüller http://d-nb.info/gnd/122522559 Kalinowski, Thomas Thomas Kalinowski http://d-nb.info/gnd/128420251 https://orcid.org/0000-0002-8444-6848 Leck, Uwe Uwe Leck http://d-nb.info/gnd/123092418 Hartmann, Sven Sven Hartmann http://d-nb.info/gnd/118124978 Roberts, Ian T. Ian T. Roberts University of Rostock, Institute of Mathematics 2005 510 Mathematics 2005 en https://purl.uni-rostock.de/rosdok/id00005898 urn:nbn:de:gbv:28-rosdok_id00005898-4 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 =: λ = λ(n) < 2. For λ ≥ 2 our problem is equivalent to the (6,3)-problem of Ruzsa and Szemerédi, and by a result of theirs it follows that min w(A) = w2n^2/2 + o(n^2).