  A (unit) disk graph is the intersection graph of closed (unit) disks in the
plane. Almost three decades ago, an elegant polynomial-time algorithm was found
for \textsc{Maximum Clique} on unit disk graphs [Clark, Colbourn, Johnson;
Discrete Mathematics '90]. Since then, it has been an intriguing open question
whether or not tractability can be extended to general disk graphs. We show
that the disjoint union of two odd cycles is never the complement of a disk
graph nor of a unit (3-dimensional) ball graph. From that fact and existing
results, we derive a simple QPTAS and a subexponential algorithm running in
time $2^{\tilde{O}(n^{2/3})}$ for \textsc{Maximum Clique} on disk and unit ball
graphs. We then obtain a randomized EPTAS for computing the independence number
on graphs having no disjoint union of two odd cycles as an induced subgraph,
bounded VC-dimension, and linear independence number. This, in combination with
our structural results, yields a randomized EPTAS for \textsc{Max Clique} on
disk and unit ball graphs. \textsc{Max Clique} on unit ball graphs is
equivalent to finding, given a collection of points in $\mathbb R^3$, a maximum
subset of points with diameter at most some fixed value. In stark contrast,
\textsc{Maximum Clique} on ball graphs and unit $4$-dimensional ball graphs, as
well as intersection graphs of filled ellipses (even close to unit disks) or
filled triangles is unlikely to have such algorithms. Indeed, we show that, for
all those problems, there is a constant ratio of approximation which cannot be
attained even in time $2^{n^{1-\varepsilon}}$, unless the Exponential Time
Hypothesis fails.
