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                * Princeton Discrete Math Seminar *
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Speaker: Dmitrii Zakharov (MIT)

Thursday 6th November, 3:00 in Fine Hall 224.

Title: A configuration of point-line pairs with large minimal distance

Say we have a collection of n points in the unit square and each point has a
line through it. Let delta be the minimal distance from a point in the
collection to a line through another point. What is the largest possible delta
among all possible collections of points and lines? It is an exercise to show
that n^{-1} < delta < n^{-1/2} but improving either of these bounds is not so
easy. In 2024, Cohen, Pohoata and myself improved the upper bound to delta <
n^{-2/3}. In this talk, I'll present a new simple construction showing that
delta > n^{c-1} for some constant c>0. Time permitting, I'll also talk a bit
about analogues of this problem over finite fields and in higher dimensions and,
in particular, a new upper bound in three dimensions.

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