What was already known
Erdős asked whether every such set has a point determining (1−o(1))n distinct distances. A formulation that also forbids three collinear points is stronger.
AI involvement
Feng and coauthors report that Aletheia found the coordinate-axis construction autonomously. Human editors retained its core logic, corrected the presentation and rejected the separate variant.
The project classifies this as an autonomously found partial solution; human editors preserved the mathematical core of the coordinate-axis construction.
Importance
Answers the no-four-cocircular version negatively with an explicit family.
Limitations
The construction allows collinear points and does not settle the no-three-collinear variant. The authors rejected the model’s proposed answer to that variant; the individual expert reviewers are not named.
Technical verification
The project reports careful domain-expert vetting and prints a complete elementary count yielding the strict 3n/4 bound. The reported human review applies only to the retained no-four-cocircular counterexample.
Formal checking
The project reports careful domain-expert vetting and prints a complete elementary count yielding the strict 3n/4 bound. The reported human review applies only to the retained no-four-cocircular counterexample.
- Checker
- Feng et al.'s internal domain experts and paper authors
- Scope
- Coordinate-axis point sets with no four cocircular have at most about 3n/4 distances from each point
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
The human-edited §3.1 proof explains the coordinate-axis arrangement and distance bound and explicitly separates the failed no-three-collinear attempt.
Correction and publication history
Answers the no-four-cocircular version negatively with an explicit family.
Read this published revision