Weak colourings of Kirkman triple systems

The article presents a comprehensive study of weak colourings in Kirkman triple systems, a specific type of balanced incomplete block design. It begins by establishing the necessary background on Steiner triple systems and their chromatic properties, highlighting the known results on the existence and chromatic numbers of these systems. The main focus is on Kirkman triple systems, which are resolvable Steiner triple systems, and their colouring behaviours. The article introduces the concept of rainbow colourings and presents several constructions for Kirkman triple systems with specific chromatic numbers. Notably, it proves the existence of 3-chromatic Kirkman triple systems for all admissible orders and provides an infinite family of 4-chromatic Kirkman triple systems. Moreover, it establishes the existence of Kirkman triple systems with an arbitrary chromatic number, generalizing previous results. The article also explores the relationship between the colouring of Kirkman triple systems and their underlying quadruple systems, presenting several theorems and examples that illustrate this connection. Additionally, it discusses open problems and conjectures, inviting further research in this area. The detailed proofs and examples make this article an essential read for anyone interested in the colouring properties of combinatorial designs.