What was already known
Erdős asked in 1957 whether every sufficiently large planar point set has even one pin determining n^(1−ε) distances. The manuscript strengthens that weak pinned formulation to all but a vanishing fraction of pins.
AI involvement
OpenAI credits itself as manuscript author and states that its internal frontier model produced the results in this release. Individual human input is not detailed enough to establish task-only autonomy.
OpenAI is named as author on the manuscript, and its release attributes the results to an internal frontier model. The individual human input is not detailed enough to establish autonomy.
Importance
Resolves the weak pinned Erdős distinct-distance conjecture in a stronger almost-every-pin form.
Limitations
The result gives n^(1−ε) for each fixed ε, not a linear-in-n pinned-distance bound. The unit-distance power saving is a separate contribution.
Technical verification
OpenAI reports a Lean proof matching the selected claim.
Formal checking
The pinned OpenAI Lean scope note identifies this selected claim.
- Checker
- OpenAI-reported Lean 4 / Comparator
- Scope
- For every fixed ε>0, all but o(n) points of any n-point planar set determine at least n^(1−ε) distinct nonzero distances as n grows, uniformly over the set.
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
Few is the default when evidence for a higher level is insufficient. No named person is documented as explaining the central argument for this assessed contribution.
Correction and publication history
Primary manuscript, attribution, prior-work, and scoped checking review for non-notifying local inclusion.
Read this published revision