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Smooth graphs

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&#39;] for some finite W&#39;. 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.