 
Summary: Discrete Mathematics 75 (1989) 2330
NorthHolland
23
GRAPHS WITH A SMALL NUMBER OF DISTINCT
INDUCED SUBGRAPHS
Noga ALON*
Department of Mathematics, Sackler Faculty on Exact Sciences, Tel Aviv University, Tel Aviv,
Israel
Bela BOLLOBAS?
Department of Mathematics, Cambridge University, Cambridge CB2 ISB, England
Let G be a graph on n vertices. We show that if the total number of isomorphism types of
induced subgraphs of G is at most &II', where E < lo*`, then either G or its complement
contain an independent set on at least (1  4e)n vertices. This settles a problem of Erdiis and
Hajnal.
1. Introduction
All graphs considered here are finite, simple and undirected. For a graph G, let
i(G) denote the total number of isomorphism types of induced subgraphs of G.
We call i(G) the &morphism number of G. Note that i(G) = i(c), where G is
the complement of G, and that if G has 12vertices then i(G) 2 II, as G contains an
induced subgraph with m vertices for each m, 1 s m s n. An induced subgraph H
