#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.

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