If every neighborhood of a city is walkable, is the city walkable? The answer feels like it should be yes. It's not.
Putterman, Sawhney, and Valiant (2602.21275) construct an infinite point set with the following property: every subset contains a dense subset with no three points collinear. But the entire set cannot be partitioned into finitely many collinear-free subsets. You can always extract good structure locally. You can never impose good structure globally.
This resolves a question of Erdős. The construction is a counterexample to what feels like a natural composability principle: if the property holds locally everywhere, it should decompose globally. The set says no. Local extractability does not imply global decomposability.
The gap is between “every piece has the property” and “the whole can be divided into pieces that each have the property.” These sound synonymous. They are not. Extracting a dense collinear-free subset from any neighborhood is a selection — you get to choose which points to keep. Partitioning the whole set into collinear-free subsets is a covering — every point must be assigned somewhere, and you don't get to leave anything out. The luxury of exclusion that makes the local property easy is exactly what the global constraint removes.
This is the structure of many real problems. Any individual team in an organization can be made functional — just select the right people, exclude the bad fits. But partitioning everyone into functional teams, without excluding anyone, is a fundamentally harder problem. The failure mode isn't that the building blocks are deficient. It's that assembly into a covering has constraints that assembly into selections does not.
The mathematical depth is in the construction: the counterexample uses algebraic independence over rationals to build a set where collinear triples are dense enough to obstruct any global partition but sparse enough locally to permit extraction. The collinear triples are the wrong kind of distributed — not concentrated enough to block local repair, not rare enough to allow global decomposition. They sit in the precise regime where the local/global distinction matters.
What Erdős suspected and these authors confirmed: the passage from “every subset has it” to “the whole decomposes into it” is not free. It costs something. And the currency it demands is the right to exclude.
Based on M. Putterman, M. Sawhney, and G. Valiant, "On infinite sets with no 3 on a line" (arXiv:2602.21275, February 2026).