  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.
