We present improved lower bounds for five classical Ramsey numbers: $\mathbf{R}(3, 13)$ is increased from $60$ to $61$, $\mathbf{R}(3, 18)$ from $99$ to $100$, $\mathbf{R}(4, 13)$ from $138$ to $139$, $\mathbf{R}(4, 14)$ from $147$ to $148$, and $\mathbf{R}(4, 15)$ from $158$ to $159$. These results were achieved using AlphaEvolve, an LLM-based code mutation agent. Beyond these new results, we successfully recovered lower bounds for all Ramsey numbers known to be exact, and matched the best known lower bounds across many other cases. These include bounds for which previous work does not detail the algorithms used. Virtually all known Ramsey lower bounds are derived computationally, with bespoke search algorithms each delivering a handful of results. AlphaEvolve is a single meta-algorithm yielding search algorithms for all of our results.

翻译：我们改进了五个经典拉姆齐数的下界：$\mathbf{R}(3, 13)$ 从 $60$ 提升至 $61$，$\mathbf{R}(3, 18)$ 从 $99$ 提升至 $100$，$\mathbf{R}(4, 13)$ 从 $138$ 提升至 $139$，$\mathbf{R}(4, 14)$ 从 $147$ 提升至 $148$，以及 $\mathbf{R}(4, 15)$ 从 $158$ 提升至 $159$。这些结果是通过使用 AlphaEvolve——一个基于大语言模型的代码变异智能体——实现的。除了这些新结果外，我们还成功恢复了所有已知精确值的拉姆齐数的下界，并在许多其他情况下匹配了已知的最佳下界。这包括那些先前工作未详述所用算法的边界。几乎所有已知的拉姆齐下界都是通过计算得出的，每种定制搜索算法各自仅能产生少量结果。而 AlphaEvolve 是一个单一的元算法，能够为我们所有结果生成相应的搜索算法。