Sierksma's Conjecture

Sierksma's conjecture asserts that every set of points in , where and , has at least unordered set partitions into nonempty parts whose convex hulls have a common point. These are the partitions guaranteed by Tverberg's theorem, and the proposed lower bound is sharp (Bárány and Soberón 2018).

For a quantitative bound valid beyond prime powers, let denote this number of set partitions. Blagojević and Blagojević (2026) proved that, for and any decomposition with integers and , and a prime power with prime and an integer,

where is the ceiling function and is a factorial. This establishes exponential growth in for each fixed , while the conjectured sharp lower bound remains open.