What was already known
Erdős expected a subquadratic maximum. Earlier no-four-collinear constructions avoided ordinary triangles but lacked a quadratic ordinary-line count.
AI involvement
An internal OpenAI model supplied the elliptic-curve coset construction and proof. Alexeev, Putterman, Sawhney, Sellke and Valiant checked and explained it.
The central construction and original proof are credited to the internal model; the exact prompting history is not disclosed.
Importance
Disproves the subquadratic expectation, including the r=3 and k=4 boundary case.
Limitations
The five human coauthors report checking the model solution and explain its mechanism. No separate machine certificate or third-party verification is documented.
Technical verification
The human coauthors explicitly report verifying each model solution; this is source-reported full human verification, not a machine formalization or an AIProofWatch rerun.
Formal checking
The human coauthors explicitly report verifying each model solution; this is source-reported full human verification, not a machine formalization or an AIProofWatch rerun.
- Checker
- Alexeev, Putterman, Sawhney, Sellke and Valiant
- Scope
- Problem 960 elliptic-curve ordinary-line construction, including r=3 and k=4
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
The authors give and interpret the elliptic-curve group-law, coset bipartition, and edge-count mechanism.
Correction and publication history
Disproves the subquadratic expectation, including the r=3 and k=4 boundary case.
Read this published revision