On colorful Helly numbers and the growth of Tverberg numbers

Abstract

In this paper, we obtain new upper and lower bounds on the colorful Helly number and the Tverberg number in an abstract convexity space with Radon number $r$. We prove an upper bound of $(r-1)2^r$ on the colorful Helly number, which is a factor $O(r)$ far from the lower bound, $2^{r-1}-1$. The best previous bound, by Holmsen and Lee (2021), was $r^{r^{\log r}}$. As a consequence, we obtain improved quantitative bounds for fractional Helly numbers, the selection lemma, weak $\varepsilon$-nets, and the $(p,q)$-theorem, in abstract convexity spaces. Furthermore, using the improved colorful Helly bound, we prove that the Tverberg number $r_k$ is at most $O(r^{\lceil \log_2 r \rceil})k$. The best previous bound, by Pálvölgyi (2022), was $r^{r^{r^{\log r}}}k$. We also study the $t$-wise Tverberg number $r_{k,t}$, which is the least $\ell$ for which any $\ell$ points can be divided into $k$ parts such that the convex hulls of any $t$ parts intersect. We prove the optimal bound $r_{k,t} = O_t(kr)$, in any $S_4$ separable space. In the other direction, we construct a separable space in which $r_k = Θ(r^2 k)$, while $r_{k,2}=Θ(rk)$. This proves that the weak version of Eckhoff's conjecture, which suggested that $r_k=O(rk)$ in any abstract convexity space, fails even in separable spaces. In addition, this shows that the abstract analogue of Reay's conjecture (1979), suggesting that $r_{k,2}=r_k$ in Euclidean spaces, already fails in separable convexity spaces.