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arXiv · 2609.26851

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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Charles Gretton, Tomasz Kowalski, Richard Pennifold, Cody Christopher. 2026-09-22. A New Method that can Generate Ramsey Colourings for Eight and Thirteen Colours. https://arxiv.org/abs/2609.26851

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