Lower bounds for Ramsey numbers: R(6,8)≥ 135 and R(8,10)≥ 345
Abstract
We prove the lower bounds R(6,8)≥ 135 and R(8,10)≥ 345, improving the bounds 134 and 343 listed in the April 2026 revision of Radziszowski's dynamic survey. We give explicit red/blue colorings of K134 with no red K6 or blue K8, and of K344 with no red K8 or blue K10, and verify them with two independently written exhaustive clique checkers. Starting from published colorings, we find these witnesses by adding vertices and repairing the resulting monochromatic cliques through local search. The search uses exact conflict counts, preparation moves aimed at making remaining cliques cheaper to break, and mutations followed by repair. A loss based on the largest monochromatic clique containing each edge yielded a useful intermediate state for the R(6,8) construction. Guided by the author, an AI coding agent wrote and ran the search code; we describe the methods and document the ancestry of the resulting colorings.
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BibTeX
@misc{cremer2026lower,
title = {{Lower bounds for Ramsey numbers: $\mathrmR(6,8)\ge 135$ and $\mathrmR(8,10)\ge 345$}},
author = {Fritz Cremer},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12122},
eprint = {2610.12122},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12122},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Lower bounds for Ramsey numbers: R(6,8)≥ 135 and R(8,10)≥ 345
AU - Fritz Cremer
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12122
UR - https://arxiv.org/abs/2610.12122
AB - We prove the lower bounds R(6,8)≥ 135 and R(8,10)≥ 345, improving the bounds 134 and 343 listed in the April 2026 revision of Radziszowski's dynamic survey. We give explicit red/blue colorings of K_134 with no red K_6 or blue K_8, and of K_344 with no red K_8 or blue K_10, and verify them with two independently written exhaustive clique checkers. Starting from published colorings, we find these witnesses by adding vertices and repairing the resulting monochromatic cliques through local search. The search uses exact conflict counts, preparation moves aimed at making remaining cliques cheaper to break, and mutations followed by repair. A loss based on the largest monochromatic clique containing each edge yielded a useful intermediate state for the R(6,8) construction. Guided by the author, an AI coding agent wrote and ran the search code; we describe the methods and document the ancestry of the resulting colorings.
ER -