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

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.

8 pages, 1 figure, 1 table