Year: 2026
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
Author: Steiner, Raphael
Institution: ETH Zurich
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7172407383