The random Borsuk graph places n points uniformly on the d-dimensional sphere and connects two points if their angular distance exceeds a threshold. The chromatic number — the minimum colors needed so no two connected vertices share a color — depends on how many edges the random process creates.
The expected behavior, by analogy with Erdős–Rényi random graphs, is that the threshold for k-colorability occurs when the average degree is logarithmic in n. More vertices require more connections before the coloring constraint binds. This is the generic random graph story: thresholds scale with log n.
For the Borsuk graph with k from 2 to d, the threshold is constant (arXiv:2603.05467). The transition from k-colorable to requiring more than k colors occurs at a fixed average degree that does not grow with n. Add more points to the sphere, and the critical connection density stays the same. The geometry of the sphere imposes a structural constraint that the generic random graph model does not have.
The mechanism is the sphere's topology. On a sphere, points at large angular separation are geometrically constrained: there are only so many directions, and distant points must be near the antipode. This constraint creates a fixed chromatic structure that is independent of how finely the sphere is sampled. Adding more points does not change the underlying geometric obstruction to coloring — it just samples it more densely.
This is a separation between geometric and combinatorial random graphs. In the combinatorial world (Erdős–Rényi), adding vertices always dilutes the constraint and the threshold scales upward. In the geometric world (Borsuk), the constraint is baked into the space and does not dilute. The threshold is a property of the sphere, not of the number of points on it.