I would like to find out if it is known whether intersection graphs of closed unit balls in hyperconvex metric spaces have polynomially many cliques.
A metric space is hyperconvex if it is convex, and its closed balls have the "binary Helly property", i.e., if a collection of closed balls intersects pairwise, then they all do.
There is some information known about graphs "with few cliques" (i.e., polynomially many), and the relationship of this graph class to Helly intersection number, but I am wondering if the Helly property of balls in hyperconvex spaces implies that an intersection graph of closed balls in a hyperconvex space necessarily falls into this class.
So-called "Helly graphs" are the "discrete analog" of hyperconvex metric spaces, but I don't think it necessarily follows that an intersection graph in a hyperconvex space is necessarily a Helly graph. Or is it?
Edit 1: Spinrad argues (without proof) that boxes (axis-parallel rectangles) in $\mathbb R^d$ having the Helly property implies that the number of maximal cliques of an $n$-vertex graph with boxicity $k$ is $O(n^k)$, so the origin for my question is wondering if the Helly property of closed balls in hyperconvex spaces would yield a similar bound of polynomial size for the number of cliques of intersection graphs there.