## On the number of certain subgraphs of graphs without large cliques and independent subsets

A graph G=<V,E> without cliques or independent subsets of size |V| is called non-trivial. We say that G= <V,E> is almost-smooth iff it is isomorphic to G[V-W] whenever W is a subset of V with |W| < |V|. Given a graph G= <V,E> denote by I(G) the set of all isomorphism classes of induced subgraphs of cardinality |V|. It is shown that
• |I(G)|>=2^omega for each non-trivial graph G=<omega_1,E>.
• under principle Diamond^+ there is a non-trivial graph G=<omega_1,E> with |I(G)|=omega_1.
• the existence of a non-trivial, almost-smooth graph on omega_1 is consistent with different set-theoretical assumptions.
• under principle Diamond^+ there exists a family F of countable subsets of omega_1 which is non-trivial in a certain sense, and which is isomorphic to {B \in F: B \subset A} whenever A \subset omega_1 is an uncountable set.