  We prove algorithmic and hardness results for the problem of finding the
largest set of a fixed diameter in the Euclidean space. In particular, we prove
that if $A^*$ is the largest subset of diameter $r$ of $n$ points in the
Euclidean space, then for every $\epsilon>0$ there exists a polynomial time
algorithm that outputs a set $B$ of size at least $|A^*|$ and of diameter at
most $r(\sqrt{2}+\epsilon)$. On the hardness side, roughly speaking, we show
that unless $P=NP$ for every $\epsilon>0$ it is not possible to guarantee the
diameter $r(\sqrt{4/3}-\epsilon)$ for $B$ even if the algorithm is allowed to
output a set of size $({95\over 94}-\epsilon)^{-1}|A^*|$.
