Finding sets of heavily intersecting objects, while minimizing their size
If I understand right, you are interested in finite projective planes.  These are the maximum combinatoric set systems in which all sets intersect each other exactly once.  To briefly answer your question, you can "improve" your construction by getting up to $\Omega(n)$ sets of size $\Omega(n^{1/2})$ and still enforcing the property that any two sets intersect on exactly one element (note: this only works for certain $n$, e.g. $n$ = square of a prime).

I assume you have some unstated restriction on your sets that rules out the construction $\{1, 2\}, \{1, 3\}, \{1, 4\} ...$ in which case your sets have size two but all pairs intersect on exactly one element?