PreprintCC BY 4.0arXiv
Multicolor Ramsey numbers of odd cycles are superexponential
Abstract
In a recent breakthrough, OpenAI proved that the k-color Ramsey number of the triangle C3 grows super-exponentially, more precisely, they proved that Rk(C3)≥ kk/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≥ 1, let 𝒪p={C3,C5,…,C2p+1}. We show that Rk(𝒪p)≥ (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.
Related
BibTeX
@misc{steiner2026multicolor,
title = {{Multicolor Ramsey numbers of odd cycles are superexponential}},
author = {Raphael Steiner},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.02537},
eprint = {2608.02537},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.02537},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Multicolor Ramsey numbers of odd cycles are superexponential
AU - Raphael Steiner
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2608.02537
UR - https://arxiv.org/abs/2608.02537
AB - 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)≥ 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≥ 1, let 𝒪_p={C_3,C_5,…,C_2p+1}. We show that R_k(𝒪_p)≥ (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.
ER -