Finding the largest set of points of limited diameter Given points $p_1,\ldots,p_n$ in $\mathbb{R}^{d}$ and a distance $l$ find the
largest subset of these points such that the Euclidian distance of no
two of them exceeds $l$.

What is the complexity of this problem?

In the graph over the points which has an edge whenever the distance
of two points is at most $l$, the problem is equivalent to finding a
maximum clique. The converse may not hold, because not every graph can
be obtained this way (an example is the star $K_{1,7}$ for $d=2$). Therefore a related question is: what is known about this class
of graphs?
There's an $O(n^3\log n)$ time algorithm for the two-dimensional version of this problem in my paper with Jeff Erickson, "Iterated nearest neighbors and finding minimal polytopes", Disc. Comp. Geom. 11:321-350, 1994. Actually the paper primarily looks at the dual problem: given the number of points in the subset, find the smallest possible diameter; but it uses the problem you describe as a subroutine. At least at the time we wrote it, we didn't know anything subexponential for higher dimensions (although if the subset has only $k$ points in it the exponential part can be made dependent on $k$ rather than $n$ using techniques in the same paper).