Homogeneous coloured multipartite graphs

TL;DR

This paper classifies all countable homogeneous coloured multipartite graphs with any finite number of parts by analyzing finite forbidden configurations and their amalgamation properties, simplifying the classification process for larger numbers of parts.

Contribution

It provides a complete classification of countable homogeneous coloured multipartite graphs for any finite number of parts, reducing complexity by focusing on the quadripartite case.

Findings

Classification of countable homogeneous coloured multipartite graphs

Reduction of classification complexity for larger numbers of parts

Identification of key finite forbidden configurations

Abstract

We classify the countable homogeneous coloured multipartite graphs with any finite number of parts. By Fraisse's Theorem this amounts to classifying the families F of pairwise non-embeddable finite coloured multipartite graphs for which the class Forb(F) of multipartite graphs which forbid these is an amalgamation class. We show that once we understand such families F in the quadripartite case, things do not become any more complicated for larger numbers of parts.