Classifying unavoidable Tverberg partitions

Let T (d, r) def = (r − 1)(d + 1) + 1 be the parameter in Tverberg’s theorem. We say that a partition I of {1, 2, . . . , T (d, r)} into r parts 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 I is a Tverberg partition. We say that I is unavoidable if it occurs in every sufficiently long point sequence. In this paper we study the problem of determining which Tverberg partitions are unavoidable. We conjecture a complete characterization of the unavoidable Tverberg partitions, and we prove some cases of our conjecture for d ≤ 4. Along the way, we study the avoidability of many other geometric predicates, and we raise many open problems. Our techniques also yield a large family of T (d, r)-point sets for which the number of Tverberg partitions is exactly (r− 1)!. This lends further support for Sierksma’s conjecture on the number of Tverberg partitions.