Gabriel Nivasch

Title: Classifying unavoidable Tverberg partitions

Abstract:

Let T(d,r):=(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.

We study the problem of determining which Tverberg partitions are
unavoidable. This is an instance of a more general problem (scarcely studied
before, to our knowledge) of determining which geometric predicates 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.

Joint work with Boris Bukh and Po-Shen Loh.