 
Summary: Constructive bounds for a Ramseytype problem
Noga Alon
Michael Krivelevich
Abstract
For every fixed integers r, s satisfying 2 r < s there exists some = (r, s) > 0
for which we construct explicitly an infinite family of graphs Hr,s,n, where Hr,s,n has n
vertices, contains no clique on s vertices and every subset of at least n1
of its vertices
contains a clique of size r. The constructions are based on spectral and geometric
techniques, some properties of Finite Geometries and certain isoperimetric inequalities.
1 Introduction
The Ramsey number R(s, t) is the smallest integer n such that every graph on n vertices
contains either a clique Ks of size s or an independent set of size t. The problem of
determining or estimating the function R(s, t) received a considerable amount of attention,
see, e.g., [14] and some of its references. A more general function was first considered (for
a special case) by Erdos and Gallai in [11]. Suppose 2 r < s n are integers, and let G
be a Ksfree graph on n vertices. Let fr(G) denote the maximum cardinality of a subset of
vertices of G that contains no copy of Kr, and define, following [12], [8]:
fr,s(n) = min fr(G),
where the minimum is taken over all Ksfree graphs G on n vertices.
