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2018

Universality of vector sequences and universality of Tverberg partitions

Por, Attila

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A result of Rosenthal says that for every $q>1$ and $n \in \mathbb{N}$ there is $N \in \mathbb{N}$ such that every sequence of $N$ distinct positive numbers contains, after a suitable translation and possible multiplication by $-1$, a subsequence $a_1,\ldots,a_n$ that is either $q$-increasing (that is, $a_{i+1}>qa_i$ for all $i$) or $1/q$-decreasing ($a_{i+1}<a_i/q$ for all $i$).

One of our main theorems extends this result to vector sequences.
This theorem is then used to prove the universality theorem for Tverberg partitions which says that, for every $d$ and $r$, every long enough sequence of points in $\mathbb{R}^d$ in general position contains a subsequence of length $n$ whose Tverberg partitions are exactly the so called rainbow partitions.

Universality of vector sequences and universality of Tverberg partitions · Around

Built on

Tverberg, H.: A generalization of Radon’s theorem. J. London Math. Soc. 41 1966 123–128
1906
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Radon, J.: Mengen konvexer Körper, die einen gemeinsamen Punkt enthalten. Math. Ann. bf 83 (1921), 113–115
1921
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Erdős, P. Szekeres, G.: A combinatorial problem in geometry. Compositio Math. 2 (1935), 463–470
1935
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D A . Rosenthal, D. A.: The classification of the order indiscernibles of real closed fields and other theories, Ph.D. dissertation, University of Wisconsin, Madison, Wis., 1981. MR 2631607
1981
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Then

Bukh, B. Loh, P., Nivasch, G.: Classifying unavoidable Tverberg partitions. J. Comput. Geom. 8 (2017), 174–205
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Bukh, B., Nivasch, G.: One-sided epsilon-approximants. A journey through discrete mathematics, 343–356, Springer, Cham, 2017
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