Let $f_r(d,s_1,\ldots,s_r)$ 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 $\bigcap_{i=1}^r C_i\neq\emptyset$, 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 $f_r(d,s_1,\dots,s_r) = O(dr^2\log r \prod_{i=1}^r s_i\cdot \log(\prod_{i=1}^r s_i).$ 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 $f_r(d,s,\ldots,s)\le Poly(r,d,s)$. We answer both in the negative by showing the exponential lower bound $f_{r}(d,s,\ldots,s)> s^{r}$ 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 $F_r(d,s_1,\ldots,s_r)$ 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, $F_r(d,s,\ldots,s)$ 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.

翻译：令$f_r(d,s_1,\ldots,s_r)$表示最小整数$n$，使得每个$n$点集$P\subseteq\mathbb{R}^d$都存在一个划分$P=P_1\cup\cdots\cup P_r$，其满足如下性质：对于任意选择的$s_i$-凸集$C_i\supseteq P_i$ $(i\in[r])$，必然有$\bigcap_{i=1}^r C_i\neq\emptyset$，其中$s_i$-凸集指$s_i$个凸集的并。Alon与Smorodinsky最近的一项突破性工作建立了一般上界$f_r(d,s_1,\dots,s_r) = O(dr^2\log r \prod_{i=1}^r s_i\cdot \log(\prod_{i=1}^r s_i))$。特别地，当$r=2$时，这解决了Kalai自1970年代提出的问题。他们进一步提出了两个特别引人关注的问题：$f_{2}(2,s,s)$是否可以从$O(s^2\log s)$改进为$O(s)$，以及$f_r(d,s,\ldots,s)\le Poly(r,d,s)$是否成立。我们通过证明对于任意$r\ge 2$、$s\ge 1$和$d\ge 2r-2$，存在指数下界$f_{r}(d,s,\ldots,s)> s^{r}$，从而对这两个问题给出了否定回答。该下界在$s$充分大时与上界仅相差一个$\log{s}$因子。我们的构造结合了一个扇贝形的平面配置与高维环面$(\mathbb{S}^1)^{r-2}$上正则$s$边形的直积。或许令人惊讶的是，如果我们额外要求每个块内的$s_i$个凸集两两不交，情况会发生显著变化。令$F_r(d,s_1,\ldots,s_r)$表示此不交并变体的极值函数。我们证明：(1) 通过将其与平面上一个合适的线分离函数相联系，得到$F_{2}(2,s,s)=O(s\log s)$；(2) 当$s$较大时，$F_r(d,s,\ldots,s)$分别可以被$O_{r,d}(s^{(1-\frac{1}{2^{d}(d+1)})r+1})$和$O_{d}(r^{3}\log r\cdot s^{2d+3})$界定。这建立在一个几何障碍与超图Turán数之间的新颖联系之上，特别是Erdős盒子问题的一个变体。