Qualitative representations of chromatic algebras

TL;DR

This paper explores edge colourings of complete graphs with specific configuration constraints motivated by qualitative reasoning, linking combinatorial properties to the qualitative representability of nonassociative relation algebras called chromatic.

Contribution

It introduces a new class of edge colourings that require certain triangles to be present, connecting combinatorial graph properties with the qualitative representability of chromatic relation algebras.

Findings

Characterization of edge colourings with prescribed triangle configurations

Connection between graph colourings and qualitative representability of nonassociative relation algebras

New insights into combinatorial properties relevant to qualitative reasoning

Abstract

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.