Question

For a first-order theory $T$ and cardinals $\kappa < \lambda$, we say that $M$ is a $(\kappa,\lambda)$-model if it is of size $\lambda$ and has a definable (with parameters) subset of size $\kappa$.

1) Let $T$ be the theory of the countable random graph. Which $(\kappa,\lambda)$-models does it admit?

2) For an arbitrary $T$, what are the sufficient conditions for the existence of $(\kappa,\lambda)$ models for some $\kappa < \lambda$? This is not a question about transfer from some $(\kappa,\lambda)$ to a different $(\kappa',\lambda')$, there are quite a few theorems there. What I am asking for is some kind of a non-structure theorem, (apart from having a Vaughtian pair).

Answer 1

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 $K_{n}$-free graph, and $\kappa \leq \lambda$ are infinite cardinals, then the following are equivalent: (1) $\lambda \leq 2^{\kappa}$; (2) there is a graph $G^{\prime}$ elementarily equivalent to G of cardinality λ and a vertex $v\in V(G^{\prime})$ for which |Δ(v)|=κ; (3) there is a graph $G^{\prime}$ elementarily equivalent to G of cardinality λ and a vertex $v\in V(G^{\prime})$ for which |Δ′(v)|=κ. (Here Δ(v) is the set of neighbors of v in G∗, and Δ′(v) is its complement.)