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: There can be four points on a line. All this says is that there exist four distinct points no three of which are collinear. There might also be other points.
EDIT: Let's try thinking of this algorithmically:
Let $P$ be the set of points.
Step 0. Make a list of all 4-element subsets of $P$. Label them $S_1, S_2,S_3,\ldots,S_n$ (it's a finite list since $P$ is finite).
Step 1. Look at $S_1$. This consists of four points. If no three of these four points are collinear, then we win, i.e. property iii holds.
Step 2. Otherwise, look at $S_2$. This consists of four points. If no three of these four points are collinear, then we win, i.e. property iii holds.
Step 3. Otherwise, look at $S_3$. This consists of four points. If no three of these four points are collinear, then we win, i.e. property iii holds.
...
Step n. Otherwise, look at $S_n$. This consists of four points. If no three of these four points are collinear, then we win, i.e. property iii holds.
Step n+1. Otherwise, we lose, i.e. property iii fails.
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 $1,2,4,7$.
