  We study the complexity of Maximum Clique in intersection graphs of convex
objects in the plane. On the algorithmic side, we extend the polynomial-time
algorithm for unit disks [Clark '90, Raghavan and Spinrad '03] to translates of
any fixed convex set. We also generalize the efficient polynomial-time
approximation scheme (EPTAS) and subexponential algorithm for disks [Bonnet et
al. '18, Bonamy et al. '18] to homothets of a fixed centrally symmetric convex
set. The main open question on that topic is the complexity of Maximum Clique
in disk graphs. It is not known whether this problem is NP-hard. We observe
that, so far, all the hardness proofs for Maximum Clique in intersection graph
classes $\mathcal I$ follow the same road. They show that, for every graph $G$
of a large-enough class $\mathcal C$, the complement of an even subdivision of
$G$ belongs to the intersection class $\mathcal I$. Then they conclude invoking
the hardness of Maximum Independent Set on the class $\mathcal C$, and the fact
that the even subdivision preserves that hardness. However there is a strong
evidence that this approach cannot work for disk graphs [Bonnet et al. '18]. We
suggest a new approach, based on a problem that we dub Max Interval Permutation
Avoidance, which we prove unlikely to have a subexponential-time approximation
scheme. We transfer that hardness to Maximum Clique in intersection graphs of
objects which can be either half-planes (or unit disks) or axis-parallel
rectangles. That problem is not amenable to the previous approach. We hope that
a scaled down (merely NP-hard) variant of Max Interval Permutation Avoidance
could help making progress on the disk case, for instance by showing the
NP-hardness for (convex) pseudo-disks.
