On the number of non-isomorphic subgraphs

Let {Mathematical expression} be the family of graphs on ω₁ without cliques or independent subsets of size w ₁. We prove that (a) it is consistent with CH that every Gε {Mathematical expression} has 2^ω many pairwise non-isomorphic subgraphs, (b) the following proposition holds in L: (*)there is a Gε {Mathematical expression} such that for each partition (A, B) of ω₁ either G≅G[A] or G≅G[B], (c) the failure of (*) is consistent with ZFC.