Inter-city bus services

Question

The ministry of transportation of Lampukistan just made the following announcement: From Jul 1st, the country-wide inter-city bus service will be organized by the ministry.

The service will be organized in the following way:

• Every city is to be connected to every other city by exactly one line.
• Every line will serve exactly three cities.

The service will have more than one line. At least how many cities does Lampukistan have?

Bonus points if you give a formula for all possible numbers of cities.

Answer 1

The bus service map is a Fano projective plane.

Each pair of cities has one line through it, and any two lines also go through exactly one common city. Furthermore, there are three cities on each line, and each city is on three lines.

In total, there are seven cities and seven lines in the system.