OpenAI刚证完三角形Ramsey数超指数，这篇直接用类似构造推广到所有奇圈，更妙的是证明是ChatGPT 5.6 Pro自动找出来的，值得看看。
本文改进了OpenAI关于三角形多色Ramsey数R_k(C_3)≥k^{k/3-o(k)}的递归构造，将其推广到固定奇圈集合O_p={C_3,C_5,...,C_{2p+1}}。作者证明对每个固定p，有R_k(O_p)≥(log^{(p-1)}k)^{k/3-o(k)}，其中log^{(p-1)}为(p-1)重迭代对数。该结果直接推出每个固定奇圈的多色Ramsey数关于颜色数k都是超指数增长的。论文指出，这个证明是由ChatGPT 5.6 Pro/Sol自动发现的。
Multicolor Ramsey numbers of odd cycles are superexponential
In a recent breakthrough, OpenAI proved that the $k$-color Ramsey number of the triangle $C_3$ grows super-exponentially, more precisely, they proved that $R_k(C_3)\ge k^{k/3-o(k)}$. In this short note, we present a modification of their recursive construction that works for multicolor Ramsey numbers of fixed odd cycles. More precisely, for $p\ge 1$, let $\mathcal{O}_p=\{C_3,C_5,\ldots,C_{2p+1}\}$. We show that \[ R_k(\mathcal{O}_p)\ge (\log^{(p-1)}k)^{k/3-o(k)} \] for every fixed $p$, where $\log^{(p-1)}$ denotes the $(p-1)$-fold iterated logarithm. This immediately implies that for every fixed odd cycle, the multicolor Ramsey number is superexponential in the number of colors. The presented proof was found autonomously by ChatGPT 5.6 Pro/Sol.