Universality of vector sequences and universality of Tverberg partitions

Abstract

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 $a_1,\dots ,a_n$ that is either $q$-increasing (that is, $a_{i+1}>q{a}_i$ 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.

30 pages

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