#4. Same-Size Intersections

Published

11 Apr, 2023

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Link to Slides · Link to recording

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.