AI 中文总结
该研究提出整数规划框架，结合循环图特性改进拉姆齐数下界，提升25个$R(3,n)$下界，还确定8个新$R_C(3,n)$值，成果可独立验证复用。
AI 中文摘要
拉姆齐数$R(m,n)$是完全图的红蓝边着色中必然包含大小为$m$的蓝色团（完全子图）或大小为$n$的红色团的最小阶数。精确确定这些数极其困难，甚至要证明一个下界也需要展示一种同时避免两种团的明确着色方案。我们开发了一种用于证明此类下界的整数规划框架，将搜索范围限制在循环图上，其旋转对称性使我们能在投影距离空间中重新表述问题，将二元变量的数量从图阶数的二次级减少到线性级。我们通过系数约简强化了该投影模型，并采用分支切割算法求解，其分离例程利用了循环图的公共邻域结构，结合了启发式算法和精确最大团算法。在对顶点数最多达410的循环图进行的大量计算实验中，我们针对$24\le n\le49$且$n\neq27$的25个$R(3,n)$值，将其他方法得到的最佳下界提高了最多11个点，每个结果都有明确的图证书支持，可通过独立的精确团求解器验证。据我们所知，我们的方法还提供了第一种可复现的基于优化的程序，用于证明循环拉姆齐数$R_C(m,n)$，我们用该程序确定了$13\le n\le20$的8个新$R_C(3,n)$值。我们的框架、图证书和独立检查器作为补充材料提供，以支持独立验证和复用。
英文摘要
The Ramsey number $R(m,n)$ is the smallest order at which every red-blue edge coloring of a complete graph must contain a blue clique (a complete subgraph) of size $m$ or a red clique of size $n$. Determining these numbers exactly is extremely hard, and even certifying a lower bound requires exhibiting an explicit coloring that avoids both cliques. We develop an integer programming framework for certifying such lower bounds, restricting the search to circulant graphs, whose rotational symmetry lets us reformulate the problem in a projected distance space, reducing the number of binary variables from quadratic to linear in the graph order. We strengthen this projected model through coefficient reduction and solve it with a branch-and-cut algorithm whose separation routine exploits the common neighborhood structure of circulant graphs, combining heuristic and exact maximum-clique algorithms. In an extensive computational campaign on circulant graphs with up to 410 vertices, we improve the best lower bounds previously obtained by other methods by up to 11 points for 25 values of $R(3,n)$ with $24\le n\le49$ and $n\neq27$, each backed by an explicit graph certificate that can be independently verified with a stand-alone exact clique solver. To the best of our knowledge, our method also provides the first reproducible optimization-based procedure for certifying circulant Ramsey numbers $R_C(m,n)$, which we use to establish eight new values of $R_C(3,n)$ with $13\le n\le20$. Our framework, graph certificates, and stand-alone checker are provided as supplementary material to support independent verification and reuse.
