#4. Same-Size Intersections

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Prompts for discussion:

1. Work out the “pedestrian proof” of the nonsingularity of B.

2. Recover the De Bruijn–Erdős theorem as a special case of the generalized Fisher inequality:

   Let P be a configuration of n points in a projective plane, not all on a line. Let t be the number of lines determined by P. Then,

   • t \geqslant n, and
   • if t = n, any two lines have exactly one point of P in common. In this case, P is either a projective plane or P is a near pencil, meaning that exactly n - 1 of the points are collinear.

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Here’s the combinatorial proof of Fisher’s inequality mentioned during the discussion.