Euclidean disk graphs — the graphs formed when disks in the plane overlap — have a known product structure. They can be decomposed into a tree-like part and a grid-like part, and this decomposition enables efficient algorithms. Bläsius, Dohse, Haun, and Merker show that moving from Euclidean to hyperbolic geometry destroys this product structure entirely.
Hyperbolic uniform disk graphs with constant clique numbers fundamentally lack the grid-like component. The geometry prevents it. In Euclidean space, bounded clique number implies bounded local density, which implies the grid-like decomposition exists. In hyperbolic space, bounded clique number does not imply bounded local density — the exponential growth of area allows sparse local neighborhoods to aggregate into globally complex structures.
This is not a technical limitation of current proof methods. It's a structural impossibility rooted in the geometry itself. The same combinatorial property (bounded clique number) means completely different things in different geometries.
The implication for algorithm design is immediate: the suite of efficient algorithms that exploit product structure in Euclidean settings cannot be transferred to hyperbolic settings by analogy. The geometry doesn't just change the constants — it changes what decompositions are possible. And the property you'd check to decide whether the decomposition exists (clique number) is the wrong invariant in the wrong geometry.