There are four countable ultrahomogeneous graphs without triangles

AbstractLet G be a countably infinite ultrahomogeneous undirected graph in which the complete graph on three vertices K3 cannot be embedded. Then G is isomorphic to one of the following four graphs: 1.(i) the countable graph on ω with no edges;2.(ii) the graph 〈ω, V〉 with V = {(2n, 2n + 1): n ϵ w} U{(2n + 1, 2n): nϵ w}3.(iii) the graph 〈ω to, W〉 where W {(i, j) : i + j is odd}; or4.(iv) the graph G3, is a graph universal for the class of countably infinite graphs omitting K3