A Tverberg-type problem of Kalai: Two negative answers to questions of Alon and Smorodinsky, and the power of disjointness

Let $f_r(d,s_1,\dots ,s_r)$ denote the least integer $n$ such that every $n$-point set $P\subseteq {\mathbb{R}}^d$ admits a partition $P=P_1\cup \cdots \cup P_r$ with the property that for any choice of $s_i$-convex sets $C_i\supseteq P_i$ $(i\in [r])$ one necessarily has ${\bigcap }_{i=1}^r{C}_i\mathrm{\ne }\mathrm{\varnothing }$, where an $s_i$-convex set means a union of $s_i$ convex sets. A recent breakthrough by Alon and Smorodinsky establishes a general upper bound $f_r(d,s_1,\dots ,s_r)=O(d{r}^2\log r{\prod }_{i=1}^r{s}_i\cdot \log ({\prod }_{i=1}^r{s}_i)\mathrm{.}$ Specializing to $r=2$ resolves the problem of Kalai from the 1970s. They further singled out two particularly intriguing questions: whether $f_2(2,s,s)$ can be improved from $O(s^2\log s)$ to $O(s)$, and whether $f_r(d,s,\dots ,s)\le Poly(r,d,s)$. We answer both in the negative by showing the exponential lower bound $f_r(d,s,\dots ,s)>s^r$ for any $r\ge 2$, $s\ge 1$ and $d\ge 2r-2$, which matches the upper bound up to a multiplicative $\log s$ factor for sufficiently large $s$. Our construction combines a scalloped planar configuration with a direct product of regular $s$-gon on the high-dimensional torus $({\mathbb{S}}^1{)}^{r-2}$. Perhaps surprisingly, if we additionally require that within each block the $s_i$ convex sets are pairwise disjoint, the picture changes markedly. Let $F_r(d,s_1,\dots ,s_r)$ denote this disjoint-union variant of the extremal function. We show: (1) $F_2(2,s,s)=O(s\log s)$ by connecting it to a suitable line-separating function in the plane; (2) when $s$ is large, $F_r(d,s,\dots ,s)$ can be bounded by $O_{r,d}(s^{(1-\frac{1}{2^d(d+1)})r+1})$ and $O_d(r^3\log r\cdot s^{2d+3})$, respectively. This builds on a novel connection between the geometric obstruction and hypergraph Turán numbers, in particular, a variant of the Erdős box problem.