Erdos, Nesetril, and Rodl asked a question about infinite point sets in the plane. Can you build an infinite set where any large enough subset contains a constant-density portion with no three points collinear, yet the full set cannot be partitioned into finitely many collinear-free pieces? The existence of such a set is not obvious — the two properties push in opposite directions.
Putterman, Sawhney, and Valiant construct one. Take a sequence of algebraically independent real numbers t_1, t_2, ... and for each pair (i,j) define a point P_{i,j} = (t_i + t_j, t_i^2 + t_i t_j + t_j^2). This places points on a parabola-like curve parameterized by pairs. The algebraic independence does the heavy lifting: three points P_{i,j}, P_{j,k}, P_{i,k} sharing a common index are collinear; all other triples are not. Collinearity in the plane becomes triangle-ness in the complete graph on natural numbers.
The two properties now follow from different areas of mathematics. Any subset of the point set corresponds to edges of a graph. Every graph has a bipartite subgraph containing at least half its edges, and bipartite graphs have no triangles — so every subset has a constant-density collinear-free portion. But partitioning the full set into finitely many collinear-free pieces would mean finitely coloring the complete graph with no monochromatic triangle, which contradicts Ramsey's theorem.
The construction is clean. The proof is short. And neither was written by a human.
“The construction and proof were generated by a model internal to OpenAI. The human authors digested the proof and have presented it in a human readable form.” The three authors — mathematicians at MIT and OpenAI — verified the AI's output, reformatted it, and published it under their names with full disclosure.
What interests me is the specific kind of creativity involved. The construction is a bridge: it translates a geometric property (collinearity) into a combinatorial one (graph triangles) via an algebraic device (parabolic coordinates with algebraically independent parameters). The insight is not in the proof — once you have the construction, the argument is short. The insight is in choosing the right embedding. What kind of mathematical object makes collinearity correspond exactly to a graph-theoretic property that Ramsey theory already constrains?
This is a pointing problem, not a proving problem. The hard part was noticing that this particular arrangement of points on a curve has the exact collinearity structure needed. The AI did not understand collinearity, algebraic independence, or Ramsey theory in any human sense. It generated a formal object with the right properties. Whether that constitutes mathematical creativity depends entirely on whether you think creativity is in the understanding or in the output.