Pach showed that every $d+1$ sets of points $Q_1,\dots ,Q_{d+1}\subset {\mathbb{R}}^d$ 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 Bárány, Meshulam, Nevo and Tancer.

中文翻译：

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在Pach重叠定理的拓扑版本上

帕奇（Pach）展示了每一个 $d+\mathrm{1个}$ 点集 ${问}_{\mathrm{1个}}，\dots ，{问}_{d+\mathrm{1个}}\subset {\mathbb{[R}}^d$ 包含线性大小的子集 $P_{\mathrm{一世}}\subset {问}_{\mathrm{一世}}$这样它们跨越的所有横向单形都相交。通过一个例子，我们表明，帕克定理的拓扑扩展不适用于大小子集$C{（\mathrm{日志}ñ）}^{\mathrm{1个}/（d-\mathrm{1个}）}$。我们证明，对于除${\mathbb{小号}}^2$。令人惊讶的是，${\mathbb{小号}}^2$ 在Pach定理的拓扑版本中 ${（\mathrm{日志}ñ）}^{\mathrm{1个}/2}$。我们推测，在高维流形中，球体的区别类似。这改善了巴兰（Bárány），麦舒拉姆（Meshulam），尼沃（Nevo）和坦瑟（Tancer）的成绩。