9 research outputs found
Tverberg's theorem is 50 Years Old: A survey
This survey presents an overview of the advances around Tverberg's theorem, focusing on the last two decades. We discuss the topological, linear-algebraic, and combinatorial aspects of Tverberg's theorem and its applications. The survey contains several open problems and conjectures. © 2018 American Mathematical Society
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum
Classifying unavoidable Tverberg partitions
Let be the parameter in Tverberg's theorem, and call a partition of into parts a Tverberg type. We say that occurs in an ordered point sequence if contains a subsequence of points such that the partition of that is order-isomorphic to is a Tverberg partition. We say that 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 . Along the way, we study the avoidability of many other geometric predicates.Our techniques also yield a large family of -point sets for which the number of Tverberg partitions is exactly . This lends further support for Sierksma's conjecture on the number of Tverberg partitions.