On a topological version of Pach's overlap theorem

Pach showed that every d+1 sets of points Q1, horizontal ellipsis ,Qd+1 subset of Rd contain linearly sized subsets Pi subset of Qi 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(logn)1/(d-1). We show that this is tight in dimension 2, for all surfaces other than S2. Surprisingly, the optimal bound for S2 in the topological version of Pach's theorem is of the order (logn)1/2. We conjecture that, among higher dimensional manifolds, spheres are similarly distinguished. This improves upon the results of Barany, Meshulam, Nevo and Tancer.