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#4. Same-Size Intersections

Published

11 Apr, 2023

(Back to course page.)

Link to Slides · Link to recording


Prompts for discussion:

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

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

    Let PPP be a configuration of nnn points in a projective plane, not all on a line. Let ttt be the number of lines determined by PPP. Then,

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

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


© 2022 • Neeldhara Misra • Credits •

 

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