AI 中文总结
该研究构造了两个251阶显式循环图，证明R(4,20)≥252，改进了拉姆齐数的相关下界，为拉姆齐数研究提供了新的构造方法。
AI 中文摘要
我们构造了两个素数阶251的显式循环图，它们不含完全子图K₄，且独立数为19，由此得出R(4,20)≥252。这一结果改进了Nagda、Raghavan、Thakurta给出的R(4,20)≥237的界，以及Radziszowski动态综述中记录的长期存在的R(4,21)≥242的界。这些图是模251的两个无向五次分圆类的32元子集，与用于R(4,22)的313阶四次剩余循环图类似。不含完全子图的性质可直接验证，独立数的结论通过对一个顶点的186顶点剩余集进行位集分支定界法得到认证。
英文摘要
We exhibit two explicit circulant graphs of prime order $251$ that are $K_4$-free and have independence number $19$. Consequently \[R(4,20)\ge 252.\] These improve the bound $R(4,20)\ge 237$ given by Nagda, Raghavan, Thakurta and the long standing bound $R(4,21)\ge 242$ recorded in Radziszowski's dynamic survey. The graphs are $32$-subsets of a pair of undirected quintic cyclotomic classes modulo $251$, in analogy with the quartic-residue circulant of order $313$ used for $R(4,22)$. Clique-freeness is elementary; the independence-number claims are certified by a bitset branch-and-bound on the $186$-vertex residual of a vertex.
CommentsCode, certificates and supplementary material available at https://zenodo.org/records/21941603
