Q:

Definition of Finite Projective Plane clarification

I do not understand part iii. Why can't there be four collinear points?
The Fano plane is an example of a $3$-uniform configuration. What about configurations that are $4$-uniform? You must have $4$ points on one line.

A:

EXAMPLE:
Consider the Fano Plane: 
Condition (iii) of the definition says: there are four points no three of which lie on a line.
Now, points $1,2,3$ do lie on a line, however, consider the points $2,3,6,7$. At most two of these points lie on a line. No three of the points  $2,3,6,7$ lie on a line. Thus, condition (iii) is satisfied.
Another example would be the points $1,2,4,7$.

