A graph G on omega_1 is called <omega-smooth
if for each uncountable subset W of omega_1,
G is isomorphic to G[W-W'] for some finite W'.
We show that in various models
of ZFC if a graph G is <omega-smooth then G is
necessarily trivial, i.e, either complete or empty.
On the other hand, we prove that the existence of a non-trivial,
<omega-smooth graph is also consistent with ZFC.
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