  Let $G$ be a finite abelian group of order $n$. For any subset $B$ of $G$
with $B=-B$, the Cayley graph $G_B$ is a graph on vertex set $G$ in which $ij$
is an edge if and only if $i-j\in B.$ It was shown by Ben Green that when $G$
is a vector space over a finite field $Z/pZ$, then there is a Cayley graph
containing neither a complete subgraph nor an independent set of size more than
$clog nloglog n,$ where $c$ is an absolute constant. In this article we observe
that a modification of his arguments shows that for an arbitrary finite abelian
group of order $n$, there is a Cayley graph containing neither a complete
subgraph nor an independent set of size more than $c(omega^3(n)log omega(n)
+log nloglog n)$, where $c$ is an absolute constant and $omega(n)$ denotes the
number of distinct prime divisors of $n$.
