MR1889546 (2003e:03064) Cherlin, Gregory(1-RTG); Thomas, Simon(1-RTG) Two cardinal properties of homogeneous graphs. (English summary) J. Symbolic Logic 67 (2002), no. 1, 217–220. 03C30 (03C65 05C99)
The main result of the paper is the following theorem: If G is the Rado graph or the generic Kn-free graph, and κ≤λ are infinite cardinals, then the following are equivalent: (1) λ≤2κ; (2) there is a graph G∗ elementarily equivalent to G of cardinality λ and a vertex v∈V(G∗) for which |Δ(v)|=κ; (3) there is a graph G∗ elementarily equivalent to G of cardinality λ and a vertex v∈V(G∗) for which |Δ′(v)|=κ. (Here Δ(v) is the set of neighbors of v in G∗, and Δ′(v) is its complement.)