发表机构
Leipzig University（莱比锡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对Székely提出的欧氏空间可测与普通色数是否不同的问题，该研究通过建立相关公式，将可测色数的诸多结果提升至普通色数，并改进了相关下界，还拓展了相关定理的适用范围。
AI 中文摘要
由于强大分析技术的存在，人们已知欧氏空间的可测色数下界远优于其普通色数的下界。事实上，著名的Hadwiger-Nelson问题的突破性下界5，比可测情形的下界晚了35年以上。这就提出了一个基本问题：正如Székely在1984年所猜想的，欧氏空间的可测色数与普通色数是否不同。我们的主要结果是$\overline\alpha(\mathbb{R}^4)=m_1(\mathbb{R}^4)$和$\chi(\mathbb{R}^4)=\chi^{(m)}(\mathbb{R}^4)$，并且对于$d\ge5$，有\\[ \overline\alpha(\mathbb{Q}^d) = \overline\alpha(\mathbb{R}^d)=m_1(\mathbb{R}^d) \qquad\text{and}\qquad \chi(\mathbb{Q}^d) = \chi(\mathbb{R}^d)=\chi^{(m)}(\mathbb{R}^d). \\]我们的定理也适用于多个禁止距离$D=\{d_1,\ldots,d_t\}$，只要$d_1^2,\ldots,d_t^2 \in \mathbb{Q}$。因此，我们立即将诸多可测色数的结果提升至普通情形。我们还借此机会进一步优化新的下界。对于多个距离，Erdős询问，当$\mathbb{R}^d$具有至多$k$个禁止距离$D$时，其色数是否随$k$指数增长。根据Bukh的一个定理，对于$d \ge 4$，我们得到\\[ \sup_{|D|=k}\chi_D(\mathbb{R}^d) \ge m_1(\mathbb{R}^d)^{-k}. \\]在解决Erdős的另一个问题上取得进展，我们证明了\\[ (2+o(1))^d \le \chi(\mathbb{R}^d) \le \left(\frac{3\sqrt{3}}{4}+o(1)\right)^d. \\]我们还大幅改进了小$d\ge4$下$\chi(\mathbb{R}^d)$的下界。我们预计我们的技术可以进一步发展，这包括将我们的主定理（即对于$d\ge 4$，$\chi(\mathbb{R}^d)=\chi^{(m)}(\mathbb{R}^d)$）扩展到$d=3$，甚至可能扩展到$d=2$，以解决Hadwiger-Nelson问题。
英文摘要
Due to the availability of powerful analytic techniques, vastly superior lower bounds are known for the measurable chromatic number of Euclidean spaces compared to their ordinary chromatic number. Indeed, even the breakthrough lower bound of 5 for the famous Hadwiger-Nelson problem lagged over 35 years behind that of the measurable setting. This raises the fundamental question of whether the measurable and ordinary chromatic number of Euclidean spaces differ as conjectured by Székely in 1984. Our main result is that $\overlineα(\mathbb{R}^4)=m_1(\mathbb{R}^4)$ and $χ(\mathbb{R}^4)=χ^{(m)}(\mathbb{R}^4)$, and for $d\ge5$ that \[ \overlineα(\mathbb{Q}^d) = \overlineα(\mathbb{R}^d)=m_1(\mathbb{R}^d) \qquad\text{and}\qquad χ(\mathbb{Q}^d) = χ(\mathbb{R}^d)=χ^{(m)}(\mathbb{R}^d). \] Our theorem also holds for multiple forbidden distances $D=\{d_1,\ldots,d_t\}$ provided that $d_1^2,\ldots,d_t^2 \in \mathbb{Q}$. As a consequence, we immediately lift numerous measurable chromatic number results into the ordinary setting. We also take the opportunity to further optimize the new bounds. For multiple distances, Erdős asked whether the chromatic number of $\mathbb{R}^d$ with up to $k$ forbidden distances $D$ grows exponentially in $k$. By a theorem of Bukh, we obtain for $d \ge 4$ that \[ \sup_{|D|=k}χ_D(\mathbb{R}^d) \ge m_1(\mathbb{R}^d)^{-k}. \] Making progress on another problem of Erdős, we prove that \[ (2+o(1))^d \le χ(\mathbb{R}^d) \le \left(\frac{3\sqrt{3}}{4}+o(1)\right)^d. \] We also vastly improve the lower bounds for $χ(\mathbb{R}^d)$ for small $d\ge4$. We expect that our techniques could be developed much further. This includes the possibility of extending our main theorem that $χ(\mathbb{R}^d)=χ^{(m)}(\mathbb{R}^d)$ for $d\ge 4$ to $d=3$ or possibly even to $d=2$ to tackle the Hadwiger-Nelson problem.
Comments47 pages, 1 table
