 
Summary: Independence numbers of locally sparse graphs
and a Ramsey type problem
Noga Alon
Abstract
Let G = (V, E) be a graph on n vertices with average degree t 1 in which for every vertex
v V the induced subgraph on the set of all neighbors of v is rcolorable. We show that the
independence number of G is at least c
log (r+1)
n
t log t, for some absolute positive constant c. This
strengthens a well known result of Ajtai, Koml´os and Szemer´edi. Combining their result with
some probabilistic arguments, we prove the following Ramsey type theorem, conjectured by Erd¨os
in 1979. There exists an absolute constant c > 0 so that in every graph on n vertices in which any
set of
n vertices contains at least one edge, there is some set of
n vertices that contains
at least c
