Using your Head is Permitted

The parliament of Prepostria has n>2 members. For its various tasks, the parliament uses multiple committees, each composed of a subset of the parliament members. Due to recently passed bills, the committees have to abide by the following rules:

1. Every committee must consist of at least two members of parliament.
2. Any two distinct committees must have exactly one common member. (This is to minimise overlaps while facilitating inter-committee communication.)
3. Every member of parliament must be in at least two committees. (They cannot be too lazy.)

As a function of n, what is the maximum possible number of Prepostria's parliamentary committees?

As usual, all answers must be accompanied by proofs.

List of solvers: