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⊆ B for all ,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)n²/8+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 min w( A)=w₂n²/2+o(n²).