 
Summary: On graphs with subgraphs having large independence numbers
Noga Alon
Benny Sudakov
Abstract
Let G be a graph on n vertices in which every induced subgraph on s = log3
n vertices has
an independent set of size at least t = log n. What is the largest q = q(n) so that every such G
must contain an independent set of size at least q ? This is one of several related questions raised
by Erdos and Hajnal. We show that q(n) = (log2
n/ log log n), investigate the more general
problem obtained by changing the parameters s and t, and discuss the connection to a related
Ramseytype problem.
1 Introduction
What is the largest f = f(n) so that every graph G on n vertices in which every induced subgraph
on log2
n vertices has an independent set of size at least log n, must contain an independent set of
size at least f ? This is one of several related questions considered by Erdos and Hajnal in the late
80s. The question appears in [3], where Erdos mentions that they thought that f(n) must be at least
n1/2 , but they could not even prove that it is at least 2 log n. As a special case of our main results
here we determine the asymptotic behavior of f(n) up to a factor of log log n, showing that in fact
