 
Summary: Constructive lower bounds for offdiagonal Ramsey numbers
Noga Alon
Pavel PudlŽak
February 22, 2002
Abstract
We describe an explicit construction which, for some fixed absolute positive constant , produces,
for every integer s > 1 and all sufficiently large m, a graph on at least m
log s/ log log s
vertices
containing neither a clique of size s nor an independent set of size m.
1 Introduction
For two positive integers s and m, the Ramsey number R(s, m) is the smallest integer R so that every
graph on R vertices contains either a clique of size s or an independent set of size m. Equivalently, this
is the smallest integer R so that in every 2coloring of all edges of the complete graph on R vertices
there is either a monochromatic clique of the first color on s vertices, or a monochromatic clique of
the second color on m vertices. The fact that these numbers are finite for all s, m is a special case
of Ramsey's well known theorem (see, e.g., [10]). In one of the first applications of the probabilistic
method in combinatorics, Erdos [7] proved that R(m, m) (m2m/2). The problem of finding explicit
edge colorings yielding a similar estimate is still open, despite a considerable amount of efforts by
