Imprimitive Homogeneous Simple 3 - graphs , part 1

We prove that there is only one imprimitive homogeneous simple unstable 3-graph with finite classes such that all predicates are realised in the union of two classes. 1 Definitions and context. Definition 1.1. An n-graph is a structure (M,R1, . . . , Rn) in which each Ri is binary, irreflexive and symmetric; also, for all distinct x, y ∈ M exactly one of the Ri holds and n ≥ 2. We assume that all the relations in the language are realised in a homogeneous n-graph. For any relation P in the language of an n-graph M and any element a, P (a) denotes the set {x ∈M : P (a, x)}. In this document, we are concerned with homogeneous 3-graphs M (that is, 3-graphs homogeneous in the language L = {R,S, T}) with simple theory in which the reflexive closure of R is an equivalence relation on M with classes of size n < ω. The stable 3-graphs in this category were classified by Lachlan: Theorem 1.2 (Lachlan 1986, [Lac86]). Every stable homogeneous 3-graph is isomorphic to one of the following: 1. P∗∗ 2. Z 3. Z ′ 4. Q∗ 5. P i ∗ 6. P i[Ki m] 7. Ki m[Q i] 8. Qi[Ki m] 9. Ki m[P i] 10. Ki m ×K n 11. Ki m[K j n[K p ]] where {i, j, k} = {R,S, T} and 1 ≤ m,n, p ≤ ω. The first five items in this list are finite “sporadic” structures. We will focus on simple unstable homogeneous 3-graphs. From this it follows in particular that S and T do not define equivalence relations, as in that case M would be a stable graph. We may also suppose that between distinct R-classes B,B′ both S and T are realised.