Maximal Flat Antichains of Minimum Weight

We study maximal families A of subsets of [n] = {1, 2, . . . , n} such that A contains only pairs and triples and A 6⊆ 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 minw(A) = (2w2+w3)n 2/8+o(n2) for 3/n ≤ 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 minw(A) = w2n 2/2 + o(n2). the electronic journal of combinatorics 16 (2009), #R69 1