算术图的着色障碍
Obstructions to coloring arithmetic graphs
- Hainan Bielefeld University of Applied Sciences（海南比尔费尔德应用技术大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明算术图$B_{205}$的色数为206，否证了彩虹级联猜想，并构造了$B_{211}$的着色界，同时给出相关猜想的有限图反例。
AI中文摘要：
算术图$B_n$将满足$\max(a,b)/\gcd(a,b)\le n$的不同$a,b\in\N$相连。我们证明$\chi(B_{205})=206$，否证了$\chi(B_n)=n$对所有$n$成立的猜想，即彩虹级联猜想。证明将算术指数瓦片的任意平铺简化为周期平铺，再简化为两个有限商族，并通过精确计算排除它们。我们还利用$\Z_{104}\times\Z_2$构造了一个$208$-着色，并证明$212\le\chi(B_{211})\le213$。$211$处的下界源于素数基数平铺刚性和已发表的长度为$211$的循环对数不存在性；我们给出了所需刚性陈述的直接证明。最后，我们记录了与列表级联着色猜想和反讽装饰猜想的等价性，并推导出两者的有限图反例。满足$\chi(B_n)>n$的最小$n$为$195$或$205$；确定哪个仍待解决。
英文摘要：
The arithmetic graph $B_n$ joins distinct $a,b\in\N$ when $\max(a,b)/\gcd(a,b)\le n$. We prove $χ(B_{205})=206$, disproving the conjecture that $χ(B_n)=n$ for every $n$, equivalently the Rainbow Cascades Conjecture. The proof reduces an arbitrary tiling by the arithmetic exponent tile to a periodic tiling, then to two families of finite quotients, which are excluded using exact computations. We also construct a $208$-coloring using $\Z_{104}\times\Z_2$ and prove $212\leχ(B_{211})\le213$. The lower bound at $211$ follows from prime-cardinality tiling rigidity and the published nonexistence of a cyclic logarithm of length $211$; we give a direct proof of the required rigidity statement. Finally, we record the equivalence with the List Cascade Coloring Conjecture and the conjecture on ironic decorations, and deduce finite graph counterexamples to both. The least $n$ with $χ(B_n)>n$ is either $195$ or $205$; determining which remains open.
