Definition of Finite Projective Plane clarification

Question

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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.

Answer 1

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 $1,2,4,7$.

Answer 2

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.