The set-system of Theorem 1.2 yields new families of explicit Ramsey-graphs.

The constant *c* given in [6] 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* [4]. 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*=*p*_{1}*p*_{2}....*p*_{r}, where *p*_{i} is the
*i*^{th} 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 *C*_{i} of
vertices in color *i*. Then the sets, corresponding to the
vertices of *C*_{i}, give a family of
sets, such that the size
of each set is divisible with *p*_{i}, but the size of the intersection
of any two elements of this set-system is not divisible by *p*_{i}.
Consequently, by Theorem 1.1,