Open Collections

Description

We show that for every $d,r,N$ positive integers there exists $n = n(d,r,N)$ such that Any sequence $p$ in $R^d$ of length $n$ has a subsequence $pâ$ of length $N$ such that Every subsequence of $pâ$ of length $T(d,r) = (r-1)(d+1)+1$ has identical Tverberg partitions, namely the ârainbowâ â partitions. A partition (or coloring) of the first $T(d,r)$ integers into $r$ parts (with $r$ colors) is called rainbow If every color appears exactly once in each of the following $r$-tuples: $(1, ⠦., r), (r, ⠦, 2r-1), (2r-1, ⠦, 3r-2), ⠦., ((d-1)r-(d-2), ⠦, d r-(d-1))$.