The number of Tverberg partitions beyond prime powers

Source abstract

Sierksma's conjecture asserts that every affine map $a\colonΔ_{(d+1)(r-1)}\longrightarrow\mathbb{R}^d$ has at least $((r-1)!)^d$ unordered Tverberg partitions into $r$ parts. Until now, lower bounds growing with $d$ were known only when $r$ is a prime power. As one of our main results we prove that for every integer $r\geq3$, every decomposition $r=kq+s$ into $q=p^α$ a prime power, $k\geq1$ and $s\geq0$, and every affine map $a$, the number $\operatorname{Tv}(a)$ of unordered Tverberg partitions satisfies $$\operatorname{Tv}(a)\ \geq\ \frac{q}{r!}\,\Big(\frac{q}{α+1}\Big)^{\big\lceil\frac{(r-1-2s)(d+1)}{2}\big\rceil}.$$