The Classification of Homogeneous Simple 3-graphs

TL;DR

This paper classifies all ultrahomogeneous complete 3-edge-coloured graphs with simple theory, extending Lachlan's stable case to include various unstable structures, providing a comprehensive taxonomy of these complex graph classes.

Contribution

It extends Lachlan's classification to include unstable ultrahomogeneous 3-graphs, detailing new structures such as primitive, imprimitive with infinite and finite classes.

Findings

Classified all stable and unstable ultrahomogeneous 3-graphs.

Identified new classes like the random 3-graph and various imprimitive structures.

Provided explicit constructions for complex graph classes.

Abstract

We classify the ultrahomogeneous complete 3-edge-coloured graphs (3-graphs) with simple theory. This extends Lachlan's result (a corollary of the Effective Classification Theorem for stable structures) classifying the stable homogeneous 3-graphs. The unstable structures in this class are: + Primitive structures: The random 3-graph $\Gamma^{i,j,k}$ + Imprimitive structures with infinite classes: * $K_m^i[\Gamma^{j,k}]$, $m\in\omega+1$ * $\Gamma^{i,j}[K_\omega^k]$ * $\mathcal B_n^{i,j}$, $n\in\omega$, $n\geq2$ * $\mathcal B^i$ + Imprimitive structures with finite classes: * $C^i(\Gamma^{j,k})$ * $\Gamma^{i,j}[K_n^k]$, $n\in\omega$ Where $\{i,j,k\}=\{R,S,T\}$, $\mathcal B_n^{i,j}$ is the random $n$-partite graph, and $\mathcal B$ is the Fra\"iss\'e limit of the class of all finite 3-graphs in which the predicate $i$ is an equivalence relation (i.e., the triangles $iij$ and $iik$ are forbidden). Finally, $C^i(\Gamma^{j,k})$ is the 3-graph obtained from the following construction: enumerate the Random Graph in predicates $j,k$ as $\{v_n:n\in\omega\}$. For each vertex $v_n$, there are two vertices, $a_n$ and $b_n$ in $C^i(\Gamma^{j,k})$ which are $i$-related. There are no more $i$-edges, and if $j(v_n,v_m)$ holds, declare $j(a_n,a_m)\wedge j(b_n,b_m)$. All other edges are of type $k$.