Radon数的超线性下界
On colorful Helly numbers and the growth of Tverberg numbers
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中文总结 AI 辅助
本研究针对抽象凸性空间，否定了Pálvölgyi提出的Radon数线性上界猜想，构造出满足超线性下界的有限凸性空间，完善了Radon数的相关理论。
中文摘要 AI 辅助
对于凸性空间X，设r(X)=r₂(X)为其Radon数，r_k(X)为第k个Radon数。已知Pálvölgyi的结果显示r_k(X)=O_{r(X)}(k)，Pálvölgyi曾询问是否存在绝对常数C，使得对每个抽象凸性空间X，都有r_k(X)≤C r(X) k。我们否定了该问题，具体构造了有限凸性空间X_d，满足r(X_d)=Θ(log d)且r_d(X_d)=Θ(d(log d)²)=Θ(d r(X_d)²)。该构造是类似离散盒凸性的空间，下界通过随机选取得到。
英文摘要
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.
发表机构
- Ariel University（阿里埃尔大学）
- University of Illinois Chicago（伊利诺伊大学芝加哥分校）
- University of Science and Technology of China（中国科学技术大学）
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