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A finite projective plane is a hypergraph in which 1. Any two edges shere exactly one node 2. There is exactly one edge containing any given pair of nodes 3. You can choose 4 nodes such that no three are contained in a common edge

I'm trying to solve an exercise to prove that for any such object, the number of nodes is $n^2+n+1$ for some n. I pretty much have a proof, which I will post as an answer. However, what I'm really looking for is a cleaner proof or way of thinking about it, as my method seems way too convoluted.

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    Not going to take the time to write up a full answer right now, but the usual method is that first you show that every line has the same number of points (call it n+1 -- n is called the order of the projective plane), then you show every point has the same number of lines, and then that this is also n+1. I guess you don't actually need *all* of that for just this, but it's true so you may as well prove it. :) Note therefore the number of lines is n^2+n+1 as well!2011-11-15
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    For an elementary and well write account of finite projective (the related finite affine planes) look at Excursions into Mathematics by A. Beck, M. Bleicher, and D. Crowe, A.K. Peters, 2000. (The Chapter Exotic Geometries, Section 9 deals with Finite Geometries.) In particular the "combinatorics" of such planes - total number of points, number of points on a line, number of lines through a point, etc. is treated.2012-01-15

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