  We propose a polynomial-time algorithm which takes as input a finite set of
points of $\mathbb R^3$ and compute, up to arbitrary precision, a maximum
subset with diameter at most $1$. More precisely, we give the first randomized
EPTAS and deterministic PTAS for Maximum Clique in unit ball graphs. Our
approximation algorithm also works on disk graphs with arbitrary radii. Almost
three decades ago, an elegant polynomial-time algorithm was found for 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. Recently, it was shown
that the disjoint union of two odd cycles is never the complement of a disk
graph [Bonnet, Giannopoulos, Kim, Rz\k{a}\.{z}ewski, Sikora; SoCG '18]. This
enabled the authors to derive a QPTAS and a subexponential algorithm for Max
Clique on disk graphs. In this paper, we improve the approximability to a
randomized EPTAS (and a deterministic PTAS). More precisely, we 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 large independence number. We then address the question of computing Max
Clique for disks in higher dimensions. We show that intersection graphs of unit
balls, like disk graphs, do not admit the complement of two odd cycles as an
induced subgraph. This, in combination with the first result, straightforwardly
yields a randomized EPTAS for Max Clique on unit ball graphs. In stark
contrast, we show that on ball and unit 4-dimensional disk graphs, Max Clique
is NP-hard and does not admit an approximation scheme even in
subexponential-time, unless the Exponential Time Hypothesis fails.
