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PreprintCC BY 4.0arXiv

A New Method that can Generate Ramsey Colourings for Eight and Thirteen Colours

  • Charles Gretton
  • Tomasz Kowalski
  • Richard Pennifold
  • Cody Christopher
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Abstract

We study representations for relation algebras corresponding to certain edge colourings of complete graphs. Previously suitable colourings were obtained for the number of colours n up to $2000$, with two exceptions: n = 8 and n = 13. Using a method that we have called the FBF (Fusion Beautiful Fusions) method, we find colourings for $8$ and $13$ colours. Our method is a new guess-and-check approach that we describe as an adaptation of Comer's finite-field method. Using FBF we construct novel colourings that are non-isomorphic to existing published colourings for 5, 6, 7, 9, 10, 11 and 12 colours.

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Cite

BibTeXRIS
BibTeX
@misc{gretton2026method,
  title = {{A New Method that can Generate Ramsey Colourings for Eight and Thirteen Colours}},
  author = {Charles Gretton and Tomasz Kowalski and Richard Pennifold and Cody Christopher},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.26851},
  eprint = {2609.26851},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.26851},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A New Method that can Generate Ramsey Colourings for Eight and Thirteen Colours
AU  - Charles Gretton
AU  - Tomasz Kowalski
AU  - Richard Pennifold
AU  - Cody Christopher
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2609.26851
UR  - https://arxiv.org/abs/2609.26851
AB  - We study representations for relation algebras corresponding to certain edge colourings of complete graphs. Previously suitable colourings were obtained for the number of colours n up to $2000$, with two exceptions: n = 8 and n = 13. Using a method that we have called the FBF (Fusion Beautiful Fusions) method, we find colourings for $8$ and $13$ colours. Our method is a new guess-and-check approach that we describe as an adaptation of Comer's finite-field method. Using FBF we construct novel colourings that are non-isomorphic to existing published colourings for 5, 6, 7, 9, 10, 11 and 12 colours.
ER  -