What was already known
Known coloring-hardness reductions did not establish the claimed arbitrary fixed gap below one third under a three-colorability promise in the manuscript’s account.
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
Strengthens the hardness of approximating coloring on three-colorable graphs and implies that finding any fixed c-coloring, c≥3, is NP-hard under this promise.
Limitations
The formalized statement concerns the independent-set gap; the c-coloring consequence follows from it. OpenAI reports the Lean check; it was not rerun here.
Technical verification
OpenAI reports a Lean proof matching the selected claim.
Formal checking
The pinned Lean note covers the deterministic 3SAT reduction to three-colorable graphs versus graphs with independent-set density below δ.
- Checker
- OpenAI-reported Lean 4 / Comparator
- Scope
- For every fixed 0<δ<1/3, it is NP-hard to distinguish three-colorable finite simple graphs from graphs whose largest independent set has fewer than δ times the vertices.
- 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