Summary: Constructive bounds for a Ramsey-type problem
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.
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.,  and some of its references. A more general function was first considered (for
a special case) by Erdos and Gallai in . Suppose 2 r < s n are integers, and let G
be a Ks-free 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 , :
fr,s(n) = min fr(G),
where the minimum is taken over all Ks-free graphs G on n vertices.