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On a topological version of Pach's overlap theorem Bukh, Boris


Pach showed that every d+1 sets of points Q_1,..,Q_{d+1} in R^d contain linearly-sized subsets P_i in 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 S^2. Surprisingly, the optimal bound for S^2 is (log n)^{1/2}. This improves upon results of Barany, Meshulam, Nevo, Tancer. Joint work with Alfredo Hubard.

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