The transversal ratio of a polytope is the minimum fraction of its vertices you must select to touch every face. In a cube, for instance, you can touch all six faces by picking just two opposite vertices — a small fraction of the eight total. The ratio is small. Structure creates efficiency.

In five or more dimensions, this efficiency can be destroyed. There exist infinite families of polytopes whose transversal ratio approaches 1 as the number of vertices grows. To touch every face, you must select nearly every vertex. Almost no vertex is redundant.

The implication is immediate and dramatic: the weak chromatic number — the minimum colors needed so that no face is monochromatic — becomes unbounded. In low dimensions, a fixed number of colors always suffices. In high dimensions, no fixed palette works. You always need more.

What makes this structural rather than merely combinatorial: in low dimensions, geometry guarantees that many vertices are “covered” by the selection of a few. Each selected vertex touches multiple faces, and the faces overlap enough that coverage cascades. In high dimensions, this cascade breaks down. Each face becomes so specific — defined by so many vertices — that touching it requires selecting at least one of its particular vertices. The faces become picky.

The through-claim: redundancy is a property of dimension, not of quantity. A polytope can have millions of vertices and still have a transversal ratio near zero — if the dimension is low enough. Or it can have the same millions and require nearly all of them — if the dimension is high enough. The number of elements doesn’t determine how much you can delegate. The space they live in does.