AI 中文总结
本文通过构造显式有理格和解析方法，将欧几里得空间色数上界改进为R^4≤43、R^5≤132等，反驳了相关猜想，并首次给出R^10低于3^n的界。
AI 中文摘要
对$\R^n$的一种着色，若没有两个同色点的距离落在禁止距离区间$[1,\ell]$内，则称该着色对禁止距离区间$[1,\ell]$是恰当的；所需的最少颜色数记为$\chi(\R^n,[1,\ell])$，其中$\ell=1$时即为Nelson–Hadwiger问题的经典色数$\chi(\R^n)$。我们证明了新的上界$\chi(\R^4)\le43$，$\chi(\R^5)\le132$，$\chi(\R^7)\le1029$，$\chi(\R^9)\le7203$，$\chi(\R^{10})\le45619$，改进了先前已知的$49$、$140$、$1372$、$17253$和$3^{10}$；特别地，这反驳了Arman、Bondarenko、Prymak和Radchenko关于$49$和$140$在$\R^4$和$\R^5$的所有格着色中为最优的猜想。前四个界来自显式的有理格——$\R^4$中的一个Eisenstein格、$\R^5$中一个一般位置的格，以及$\R^7$和$\R^9$中Eisenstein着色$E_6^*/343$和$E_8/2401$的层叠——并且每个界都通过一个验证协议被约化为显式写出的有理数之间的有限不等式列表，这些不等式在精确算术中检验。第五个界是解析的：我们证明对每个Eisenstein格$\Lambda$，$(3+\omega)\Lambda$的同色胞之间的距离等于$\sqrt{7/3}\\,\lambda_1(\Lambda)$，这给出了所有已知的$7^{n/2}$色着色的精确宽度，以及正交积宽度的一个乘积规则$\sum_i1/d_i^2\le1$；它们一起产生了$45619=2401\cdot19$，这是$\R^{10}$中第一个低于$3^n$的界，以及$\chi(\R^{25})\le4\cdot7^{12}$和$\chi(\R^{26})\le19\cdot7^{12}$。我们还证明了$E_8$中指数低于$2401$的任何子格都不能定义恰当着色。所有代码、精确证书和数据都是开放的。
英文摘要
A coloring of $\R^n$ is \emph{proper for the forbidden distance segment} $[1,\ell]$ if no two points of the same color are at a distance from $[1,\ell]$; the minimum number of colors is $χ(\R^n,[1,\ell])$, and $\ell=1$ gives the classical chromatic number $χ(\R^n)$ of the Nelson--Hadwiger problem. We prove the new upper bounds $χ(\R^4)\le43$, $χ(\R^5)\le132$, $χ(\R^7)\le1029$, $χ(\R^9)\le7203$, $χ(\R^{10})\le45619$, improving the previously known $49$, $140$, $1372$, $17253$ and $3^{10}$; in particular, this refutes the conjecture of Arman, Bondarenko, Prymak and Radchenko that $49$ and $140$ are optimal among all lattice colorings of $\R^4$ and $\R^5$. The first four bounds come from explicit rational lattices --- an Eisenstein lattice in $\R^4$, a lattice in general position in $\R^5$, and laminations of the Eisenstein colorings $E_6^*/343$ and $E_8/2401$ in $\R^7$ and $\R^9$ --- and each is reduced, by one verification protocol, to a finite list of inequalities between explicitly written rational numbers checked in exact arithmetic. The fifth bound is analytic: we prove that for every Eisenstein lattice $Λ$ the distance between same-colored cells of $(3+ω)Λ$ equals $\sqrt{7/3}\,λ_1(Λ)$, which gives the exact widths of all known colorings with $7^{n/2}$ colors, and a product rule $\sum_i1/d_i^2\le1$ for the widths of orthogonal products; together they yield $45619=2401\cdot19$, the first bound in $\R^{10}$ below $3^n$, as well as $χ(\R^{25})\le4\cdot7^{12}$ and $χ(\R^{26})\le19\cdot7^{12}$. We also show that no sublattice of $E_8$ of index below $2401$ defines a proper coloring. All code, exact certificates and data are open.
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