  Let $T(d,r) = (r-1)(d+1)+1$ be the parameter in Tverberg's 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 $d\le 4$. 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 Sierksma's conjecture on the number of Tverberg partitions.
