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I'm attempting to solve a little exercise that my maths lecturer has given me relating to projective planes. Basically, she has asked me to show that a projective plane contains a set of 7 points and 6 lines. I was wondering how this can be done, as using the axioms that define a projective plane, I can only achieve a Fano plane which contains 7 points and 7 lines as a circular line has to be included to satisfy the axioms.

The axioms in question are:

  1. Any two distinct points lie on a unique line.
  2. Any two distinct lines meet in a unique point.
  3. There exist at least three non-collinear points.
  4. There exist at least three points on every line.

Any tips or help will be greatly appreciated!

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    $\begingroup$ aHave you considered the possibility that your lecturer may be wrong? Could there be a typo in the exercise? $\endgroup$ – bof Jun 3 '18 at 19:22
  • $\begingroup$ I wanted to know whether it was possible to solve it for six lines and seven points before asking whether the exercise contained a typo or error. But I'll admit that the thought had crossed my mind. $\endgroup$ – Maths Matador Jun 3 '18 at 19:26
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The exercise is wrong; any projective plane has as many lines as points. Let me try to give you the simplest proof from the axioms that $7$ points and $6$ lines is impossible.

There are $\binom72=21$ (unordered) pairs of points. Since each pair of points determines a line, if there are fewer then $7$ lines, then some line must be determined by more than $3$ pairs of points; i.e., some line must contain more than $3$ points. It will suffice to show that, if there are more than $3$ points on some line, then there are more than $7$ points all told.

Suppose that there are $4$ distinct points $Q_1, Q_2, Q_3, Q_4$ on the line $L.$ By Axiom 3, there is a point $P$ which is not on the line $L.$ Let $L_i$ be the line through $P$ and $Q_i.$ It's easy to see that the lines $L,L_1,L_2,L_3,L_4$ are all distinct. Each line $L_i$ must contain another point, call it $R_i,$ besides $P$ and $Q_i.$ Then the nine points $P,Q_1,Q_2,Q_3,Q_4,R_1,R_2,R_3,R_4$ are all distinct.

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