Edge colourings and qualitative representations of chromatic algebras

Conventional Ramsey-theoretic investigations for edge-colourings of complete graphs are framed around avoidance of certain configurations. Motivated by considerations arising in the field of Qualitative Reasoning, we explore edge colourings that in addition to forbidding certain triangle configurations also require others to be present. These conditions have natural combinatorial interest in their own right, but also correspond to qualitative representability of certain nonassociative relation algebras, which we will call chromatic. In an edge n-colouring of a complete graph, each triangle of edges consists of either one colour, two colours or three colours: monochromatic, dichromatic or trichromatic. This article investigates edge-colourings determined by disallowed triangle colour combinations, while also requiring specific combinations to be present. Our findings suggest that while disallowing monochromatic triangles restricts edge-coloured complete graphs within the Ramsey bound, the additional constraints raise pertinent questions about network feasibility. These considerations not only offer insights into combinatorial structures but also tie into the algebraic foundations pivotal to qualitative reasoning applied in various fields including artificial intelligence.