On the number of non-isomorphic subgraphs
  Let K be the family of graphs on omega_1 without cliques or independent subsets of size omega_1 . We prove that:   1) it is consistent with CH that every G in K has 2^{omega_1} many pairwise non-isomorphic subgraphs,   2) the following proposition holds in L: (*) there is a G in K such that for each partition (A,B) of omega_1 either G cong G[A] or G cong G[B],   3) the failure of (*) is consistent with ZFC. 
Saharon Shelah, Lajos Soukup