Let $A$ be a set with $n$ elements. Call a subset $C$ of the power set of $A$ "good" if

Each element of $C$ has at least three elements.

If $P, Q\in C$ and $P\cap Q$ has more than one element, then $P=Q$.

I've been interested in finding good upper and lower bounds of the number of good collections, but I haven't made any headway. Does anyone know of any?

**Edit:** I've gotten some good answers here that tell me that these conditions give too many collections for what I'm trying to do. I'm trying to get an asymptotic formula for the logarithm of the number of isomorphism classes of intervals in weak order of length $n$ in a Coxeter group. The weak order intervals are almost distributive lattices, and the basic underlying structure is a partially ordered set (similar to the Birkhoff representation theorem). I think the extra structure characterizing the weak order interval doesn't add anything asymptotically to the logarithm, so I think it is asymptotically $\frac {n^2}4$. A collection of this sort in addition to the partially ordered set structure would characterize the isomorphism class, but the conditions I'm giving are very loose. It turns out that the logarithm of the number of these collections is asymptotically bigger than $n^2$, so that's no good to get the result I want.

The structure characterizing the weak order interval is the inversion set of the element in the root system. A reduced word for the element is obtained by certain orderings of the inversion set, and the partial ordering arises as the relation that $x\leq y$ if $x$ comes before $y$ in every ordering of the inversion set corresponding to a reduced word. The remaining antichains are then covered by dihedral subsystems corresponding to braid moves in the words, and the ways of arranging these are what I'm trying to count. By default here we assume they are of size $2$, hence the restriction to size at least $3$. They are intersections of two dimensional subspaces with the inversion set, so if their intersection contains two elements then they are equal. There are very heavy restrictions, for example, if the entire inversion set is an antichain; in that case the inversion set is a direct sum of irreducible finite root systems, which have a very small number of isomorphism classes. I have some ideas for stronger restrictions on the collections using the classification of finite root systems, but nothing that fully characterizes the inversion set so far.

If anyone has any ideas on the more detailed problem, I'd be happy to hear them. I'm not going to invalidate the existing answers though, so I'll leave this as is.

I've done some very small computations on the actual number of isomorphism classes, as I can't think of a good way to do it with a computer. https://oeis.org/A185349. Compare to https://oeis.org/A000112.

1more comments