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arXiv · 1708.04350

On a topological version of Pach's overlap theorem

Boris Bukh·Alfredo Hubard
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Abstract

Pach showed that every $d+1$ sets of points $Q_1,\dotsc,Q_{d+1} \subset \mathbb{R}^d$ contain linearly-sized subsets $P_i\subset Q_i$ such that all the transversal simplices that they span intersect. We show, by means of an example, that a topological extension of Pach's theorem does not hold with subsets of size $C(\log n)^{1/(d-1)}$. We show that this is tight in dimension $2$, for all surfaces other than $\mathbb{S}^2$. Surprisingly, the optimal bound for $\mathbb{S}^2$ in the topological version of Pach's theorem is of the order $(\log n)^{1/2}$. We conjecture that, among higher-dimensional manifolds, spheres are similarly distinguished. This improves upon the results of B\'ar\'any, Meshulam, Nevo and Tancer.

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Boris Bukh, Alfredo Hubard. 2017-08-14. On a topological version of Pach's overlap theorem. https://doi.org/10.1112/blms.12302

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