Let $T(d,r) = (r-1)(d+1)+1$ be the parameter in Tverbergs theorem, and call a partition $mathcal I$ of ${1,2,ldots,T(d,r)}$ into $r$ parts a Tverberg type. We say that $mathcal I$ occurs in an ordered point sequence $P$ if $P$ contains a subsequence $P$ of $T(d,r)$ points such that the partition of $P$ that is order-isomorphic to $mathcal I$ is a Tverberg partition. We say that $mathcal I$ is unavoidable if it occurs in every sufficiently long point sequence. In this paper we study the problem of determining which Tverberg types are unavoidable. We conjecture a complete characterization of the unavoidable Tverberg types, and we prove some cases of our conjecture for $dle 4$. Along the way, we study the avoidability of many other geometric predicates. Our techniques also yield a large family of $T(d,r)$-point sets for which the number of Tverberg partitions is exactly $(r-1)!^d$. This lends further support for Sierksmas conjecture on the number of Tverberg partitions.
تحميل البحث