AI 中文总结
本研究通过构造明确的红蓝着色方案，结合局部搜索等方法，改进了拉姆齐数R(6,8)和R(8,10)的下界，分别提升至135和345。
AI 中文摘要
我们证明了下界$\mathrm{R}(6,8)\ge 135$和$\mathrm{R}(8,10)\ge 345$，改进了Radziszowski动态调查2026年4月修订版中列出的134和343的界。我们给出了$K_{134}$的明确红蓝着色方案，其中不含红色$K_6$或蓝色$K_8$，以及$K_{344}$的明确红蓝着色方案，其中不含红色$K_8$或蓝色$K_{10}$，并通过两个独立编写的完全团检查器对其进行了验证。从已发表的着色方案出发，我们通过添加顶点并使用局部搜索修复由此产生的单色团，找到了这些见证。该搜索使用精确的冲突计数、旨在降低剩余团破坏成本的准备操作，以及变异后再修复的方法。基于包含每条边的最大单色团的损失，为$\mathrm{R}(6,8)$的构造提供了有用的中间状态。在作者的指导下，一个AI编码代理编写并运行了搜索代码；我们描述了这些方法并记录了所得着色方案的来源。
英文摘要
We prove the lower bounds $\mathrm{R}(6,8)\ge 135$ and $\mathrm{R}(8,10)\ge 345$, improving the bounds 134 and 343 listed in the April 2026 revision of Radziszowski's dynamic survey. We give explicit red/blue colorings of $K_{134}$ with no red $K_6$ or blue $K_8$, and of $K_{344}$ with no red $K_8$ or blue $K_{10}$, and verify them with two independently written exhaustive clique checkers. Starting from published colorings, we find these witnesses by adding vertices and repairing the resulting monochromatic cliques through local search. The search uses exact conflict counts, preparation moves aimed at making remaining cliques cheaper to break, and mutations followed by repair. A loss based on the largest monochromatic clique containing each edge yielded a useful intermediate state for the $\mathrm{R}(6,8)$ construction. Guided by the author, an AI coding agent wrote and ran the search code; we describe the methods and document the ancestry of the resulting colorings.
Comments8 pages, 2 figures
