Tverberg’s theorem says that a set with sufficiently many points in $${\mathbb {R}}^d$$ can always be partitioned into m parts so that the $$(m-1)$$ -simplex is the (nerve) intersection pattern of the convex hulls of the parts. The main results of our paper demonstrate that Tverberg’s theorem is just a special case of a more general situation, where other simplicial complexes must always arise as nerve complexes, as soon as the number of points is large enough. We prove that, given a set with sufficiently many points, all trees and all cycles can also be induced by at least one partition of the point set. We also discuss how some simplicial complexes can never be achieved this way, even for arbitrarily large sets of points.

中文翻译：

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具有改变交叉模式的 Tverberg 型定理（神经）

Tverberg 定理说，在 $${\mathbb {R}}^d$$ 中有足够多点的集合总是可以划分为 m 个部分，因此 $$(m-1)$$ -simplex 是（神经）零件凸包的相交图案。我们论文的主要结果表明，特弗伯格定理只是更一般情况的一个特例，只要点的数量足够大，其他单纯复形必须总是作为神经络合物出现。我们证明，给定一个具有足够多点的集合，所有树和所有循环也可以由该点集的至少一个分区引起。我们还讨论了一些单纯复形如何永远无法通过这种方式实现，即使对于任意大的点集也是如此。