Representability of Ramsey Relation Algebras

We obtain representations for relation algebras corresponding to certain edge colourings of complete graphs. Suitable colourings are obtained for the number of colours n up to 120, with two exceptions: n = 8 and n = 13. For n > 7 it was not known whether representations exist. The method involves a theoretical framework that, although focused on relation algebras, can also be framed as a general colouring problem for complete graphs. The study investigates the limitations and possibilities of edge colourings that avoid monochromatic triangles while ensuring that every non-monochromatic triangle appears wherever it can. Through a construction based on finite fields meeting specific criteria, we demonstrate that appropriate colouring schemes are achievable, contrasting with earlier attempts that overlooked certain solutions. This work not only clarifies the conditions necessary for successful colourings but also expands on previous findings by providing insights and results for higher values of n, ultimately asserting the existence of colourings under previously unestablished parameters.