Straight outta 2018

I've placed 2018 points in a plane such that it is not possible to draw a straight line that passes through exactly two points.

If a line passes through more than two of the points, then what is the minimum number of points that it passes through?

If 3 points are placed on a plane such that it is not possible to draw a line through exactly two points, then the 3 points must be collinear.