The set-system of Theorem 1.2 yields new families of explicit Ramsey-graphs.
The constant c given in  is . Our construction yields only.
In addition to giving a novel proof of Theorem 4.1, we extend it to the case of several colors:
The existence of graphs with more than 2(t-1)/2 vertices without a complete graph or an independent set of size t is known from the famous theorem of Erdős . The probabilistic proof of that theorem immediately implies the existence of an r-coloring of the edges of the complete graph on r(t-1)/2 vertices, without a monochromatic complete graph on t vertices.
For more than two colors, no explicit Ramsey-graph constructions seem to have appeared prior to the present work. It does not seem immediate how one could modify the Frankl-Wilson construction to more than two colors.
Proof. Let m=p1p2....pr, where pi is the ith prime. Let K be a complete graph on vertex-set , where is a set-system with the properties stated in Theorem 1.2, with . We define an r-coloring of the edges of K by colors 1,2,...,r as follows: edge UV, where , has color i if
Now suppose that K contains a monochromatic complete graph Ci of
vertices in color i. Then the sets, corresponding to the
vertices of Ci, give a family of
sets, such that the size
of each set is divisible with pi, but the size of the intersection
of any two elements of this set-system is not divisible by pi.
Consequently, by Theorem 1.1,