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Index
PROJECTIVE PLANES
1. Two distinct points are contained in a unique line.
2. Two distinct lines intersect at a unique point.
3. There exist four points of which no three are incident with the same line.
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Recall that in problem 2, we defined the order of a projective plane to be some integer k, such that each line is incident with k+1 points and each point is incident with k+1 lines. In addition, the point set and the line set both have b = v = k2+k+1 elements. Such a linear space exemplifies the second case of (ii) of the de Bruijn-Erdos theorem.
To construct a projective plane, consider the following situation.
There must be at least two distinct lines l and l' in our projective plane. There must also be a point p that does not lie on either line (can you see why such a point p must exist?). Let q be any point that lies on l. When a line is drawn connnecting q and p this line must be incident with l' at a unique point (why?). The point at which q intersects l' will be called q'. It follows that we can define a mapping from l onto l' by sending point q to q', because any q' on l' will have a unique corresponding q on l, created by extending the line from q' through p to l. Similarly this is a 1-1 mapping because for any two distinct points q and r on l, the line from q through p will be different than the line from r through p. In addition q' can be mapped back to q, and r' back to r, by extending the lines of q' to p and r' to p back through to l. The points where these intersect l will obviously be q and r. Every point on l corresponds to a unique point on l'. Thus the incidence of each of the two lines is equivalent. Note that for any two lines in a space such that p does not lie on either of these lines, we must follow the same mapping. Note that l' is simply any line different from l. Thus it follows that every line will be incident with the same number of lines as l is. Let this number be k+1, and it follows from the fact that all k+1 lines will meet another at a given point and that the incidence of each point is also k+1.
Let's look at the creation of the Fano plane to make this appear a bit simpler:
Any finite projective plane can be created in this manner!
Notice, also, that spaces created in this way are both point regularand line regular.
Recall from Problem 9 that if a linear space is line regular then it is point regular.
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PROBLEM 9: Using the lemma that for any point pi,
where,  denotes the number of points on a given line lj, and
 = 0 if pi is not incident with lj and = 1 if pi is incident with lj, prove that if a linear space is line regular, then it is point regular. Let k+1 be the line regularity.
Finally, using your proof, show that if a linear space has point and line regularity k+1, k > 1, and all lines meet, then b = v = k2+k+1
This is what we defined as the order of a projective plane!
Solution
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References: Batten, Polster
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