2015/05/05 by Andres Aranda, Aranda, Andres
Mathematics · #F.4.1 #FOS: Mathematics #G.2.1 #Logic (math.LO) #math.LO
paper · pdf · doi:10.48550/arxiv.1505.01188
97 pages, 26 figures
arxiv created 2015/05/05 · arxiv updated 2015/05/07
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 Γi,j,k + Imprimitive structures with infinite classes: * Kmi[Γj,k], m∈ω+1 * Γi,j[Kωk] * \mathcal Bni,j, n∈ω, n≥2 * \mathcal Bi + Imprimitive structures with finite classes: * Ci(Γj,k) * Γi,j[Knk], n∈ω Where \i,j,k\=\R,S,T\, \mathcal Bni,j is the random n-partite graph, and \mathcal B is the Fraïssé 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, Ci(Γj,k) is the 3-graph obtained from the following construction: enumerate the Random Graph in predicates j,k as \vn:n∈ω\. For each vertex vn, there are two vertices, an and bn in Ci(Γj,k) which are i-related. There are no more i-edges, and if j(vn,vm) holds, declare j(an,am)\wedge j(bn,bm). All other edges are of type k.