What was already known
Ajtai, Komlós and Szemerédi established the matching upper logarithmic power s−2. Bradač reached the polynomial exponent s−1 with lower logarithmic power 2s−4; the sharp logarithmic gap for fixed s≥6 remained.
AI involvement
OpenAI is the credited author and attributes its mathematics release to an internal model. The source does not give a proof-specific autonomy trace or name a human checker or explainer of the new argument.
The manuscripts are authored OpenAI and the cited release attributes their mathematical work to an internal model; autonomous session provenance is not given.
Importance
The proof closes the logarithmic-exponent gap across all fixed s≥6.
Limitations
The pinned Lean module reports the fixed-s≥6 bounds and exponent limit. The project was not independently compiled, and no completed human proof check is documented. The separately published s=5 case is outside this record.
Technical verification
The cited Lean module reports the fixed-s-at-least-six sharp bounds and exponent limit. The source reports formalization; the project was not compiled or independently audited here.
Formal checking
Lean proof OAI.SharpLogRamsey.main
- Checker
- Lean 4 kernel, as reported by the OpenAI math release
- Scope
- For each fixed s at least six, r(s,t) has logarithmic exponent s-2 with eventual upper and epsilon-slack lower bounds
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
No named human account of the new selected-law entropy and sparse-pair compression argument is documented. Few is the evidence default.
Correction and publication history
Initial publication of this result.
Read this published revision