S. Shelah, L. Soukup:
On the number of nonisomorphic
subgraphs
Let K be the family of graphs on omega_1 without
cliques or independent subsets of size omega_1.
We prove that

it is consistent with CH that every G from K has 2^omega_1 many
pairwise nonisomorphic subgraphs,

the following proposition holds in L:
(*) ``there is a G in K such that for each partition (A,B) of
omega_1
either G[A] or G[B] is isomorphic to G
 the failure of (*) is consistent with ZFC.
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appeared in Israel J. Math 86(1994) no 13, 349371.