BG-MO 2002 Problem 3

Given are $n^2$ points in the plane, such that no three of them are collinear, where $n \geq 4$ is the positive integer of the form $3k+1$ . What is the minimal number of connecting segments among the points, such that for each $n$ -plet of points we can find four points, which are all connected to each other?

Proposed by Alexander Ivanov and Emil Kolev