An elementary exposition to topological overlap in the plane
Amir Yehudayoff

TL;DR
This paper provides an accessible, self-contained explanation of Gromov's topological overlap theorem in the plane, including a weighted generalization using von Neumann's minimax theorem.
Contribution
It offers an elementary exposition of Gromov's argument and introduces a weighted version leveraging probability distributions and duality principles.
Findings
Elementary proof of Gromov's topological overlap in the plane
Weighted generalization using probability distributions
Application of von Neumann's minimax theorem
Abstract
The aim of this text is to provide an elementary and self-contained exposition of Gromov's argument on topological overlap (the presentation is based on Gromov's work, as well as two follow-up papers of Matousek and Wagner, and of Dotterrer, Kaufman and Wagner). We also discuss a simple generalization in which the vertices are weighted according to some probability distribution. This allows to use von Neumann's minimax theorem to deduce a dual statement.
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