发表机构
Capital Normal University; University of Science and Technology of China; Zhejiang University（首都师范大学; 中国科学技术大学; 浙江大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究凸集并的特弗伯格定理中$f_{r}(d,s_{1},\ldots,s_{r})$的界，通过证明渐近精确下界、一般下界及改进上界得到$f_r(d,s,\ldots,s)=\Theta_{d,r}(s^r\log s)$等结果，还研究了两个着色类似物并证明了完全横截定理。
AI 中文摘要
设$f_{r}(d,s_{1},\ldots,s_{r})$为最小的$N$，使得每个$N$点集$P\subseteq \mathbb{R}^{d}$都有一个$r$划分$P = P_{1}\sqcup\cdots\sqcup P_{r}$，具有如下性质：当$C_{i}\supseteq P_{i}$是至多$s_{i}$个凸集的并时，有$\bigcap_{i = 1}^{r}C_{i}\neq\emptyset$。阿隆和斯莫罗丁斯基最近的突破为此问题建立了首个有效上界$f_{r}(d,s,\ldots,s)\leq Cdr^{2}s^{r}\log r\log(es^{r})$。我们通过证明对于每个$d\geq r + 2$，$f_r(d,s,\ldots,s)\geq c(d - r + 2)s^r\log(s + 1)$得到一个渐近精确下界，表明对于每个固定的$d\geq r + 2$，$f_r(d,s,\ldots,s)=\Theta_{d,r}(s^r\log s)$。我们还证明了一般下界$f_r(d,s,\ldots,s)>s^{\min\{d,r\}}$。另一方面，我们发展了局部计数论证以表明当$r\geq d + 1$时，$f_r(d,s,\ldots,s)\leq C_{d}rs^r\log(ers^r)$且$f_r(d,s,\ldots,s)\leq C_{d}r^{d + 2}s^{d + 1}\log(ers)$，改进了阿隆和斯莫罗丁斯基的上界。我们还研究了两个着色类似物。直接的巴拉尼 - 拉曼型扩展，即从$d + 1$个颜色类中寻找$r$个不相交的彩虹集，一旦允许两个凸块就不成立。然而，我们确定了正确的着色公式并证明了一个带有定量界的完全横截定理，凯勒和斯莫罗丁斯基也独立得到了该定理。
英文摘要
Let $f_{r}(d,s_{1},\ldots,s_{r})$ be the least $N$ such that every $N$-point set $P\subseteq\mathbb{R}^{d}$ has an $r$-partition $P=P_{1}\sqcup\cdots\sqcup P_{r}$ with the following property: whenever $C_{i}\supseteq P_{i}$ is a union of at most $s_{i}$ convex sets, one has $\bigcap_{i=1}^{r}C_{i}\ne\emptyset$. A recent breakthrough of Alon and Smorodinsky proved that $f_{r}(d,s,\ldots,s)\le cdr^{2}s^{r}\log r\log(es^{r})$ for an absolute constant $c>0$. In this paper, we determine the asymptotic order in two principal ranges: $f_{2}(2,s,s)=Θ(s^{2})$, and $f_{r}(d,s,\ldots,s)=Θ_{d,r}(s^{r}\log{s})$ for every fixed $d,r$ with $d\ge r+2$. The first one determines the order of the extremal function proposed by Kalai from the 1970s. Together, the two results show a sharp dependence on the dimension: for two parts, the logarithmic factor disappears in the plane but is necessary in every fixed dimension $d\ge4$. Beyond these sharp results, when $r\ge d+1$ we improve the upper bound of Alon and Smorodinsky by proving both $f_{r}(d,s,\ldots,s)\le c_{d}rs^{r}\log(ers^{r})$ and $f_{r}(d,s,\ldots,s)\le c_{d}r^{d+2}s^{d+1}\log(ers)$ through a local Helly-type argument. We also prove $f_{r}(d,s,\ldots,s)>s^{\min\{r,d\}}$ for every $d\ge2$. Finally, we study two colored analogues. The direct Bárány--Larman-type extension, in which one seeks $r$ disjoint rainbow sets chosen from $d+1$ color classes, fails as soon as two convex pieces are allowed. Nevertheless, a different extension does hold: given sufficiently many prescribed $r$-point classes, one can split every class completely among the $r$ final parts while retaining the required intersection property.
Comments23 pages. A new result was added: $f_{2}(2,s,s)=Θ(s^{2})$
