Not evaluated
Exact crossing numbers of complete and complete bipartite graphs (family 165)
The gist
On October 6, 2026, OpenAI announced, with an unreleased internal model, two manuscripts on the crossing number of complete graphs and of complete bipartite graphs.
DiscoverIA status on October 12, 2026: not evaluated, meaning no third-party evaluation found.
AI draft, not reviewedWhat has not been reviewed?
Text written by an AI (DiscoverIA pipeline), on . The status and its sources come from the registry; only the explanatory text has not been reviewed. A reported error is corrected, then the text is reviewed.
Cite this entry
A single status snapshot since Oct 12, 2026: a citation always points to a dated snapshot.
Cite snapshot 1, Oct 12, 2026
APA citation with the dated link
BibTeX @misc, key discoveria-oai-fam-165-exact-crossing-numbers-complete-complete-r1
@misc{discoveria-oai-fam-165-exact-crossing-numbers-complete-complete-r1,
author = {{DiscoverIA}},
title = {Exact crossing numbers of complete and complete bipartite graphs (family 165)},
howpublished = {DiscoverIA registry, entry oai-fam-165-exact-crossing-numbers-complete-complete},
note = {Status “Not evaluated”, snapshot 1, recorded on October 12, 2026. New result},
year = {2026},
url = {https://discoveria.ai/entry/oai-fam-165-exact-crossing-numbers-complete-complete#releve-1},
urldate = {2026-10-12}
}Dated link opens this snapshot in the history
To understand ≈ 30 s reading time, estimated
What it changes
The crossing number of a graph is the smallest number of edge crossings obtainable when drawing it in the plane. A complete graph joins every pair of vertices; a complete bipartite graph has two groups of vertices, each joined to all those of the other group. If the manuscripts held, they would bring results on this number for these two families of graphs.
What remains unclear
The Lean (a language in which a computer checks every step of a proof) is supplied by OpenAI itself, and its tracking file carries the mention “unchecked”. No third-party verification is recorded. It is not known what the manuscripts establish about this number.
For experts
Sources, method, key facts, status history, report an error
Status
Not evaluated
Recorded on · github.com · 165.md
Step 1 of 4 on the scale: not evaluated. Next step: under review.
- Announced
- Recordednot evaluated
- nextTo follow
What this status means
Source reading : Source read word for word (level 3 of 3)
The result is declared neither right nor wrong.
To move to “Under review”: a public review under way. To “Peer reviewed”: a journal publication or a review by named experts.
Sources
- Main sourcegithub.com · CONTENTS.mdlink only, reading not recorded
- Repositorygithub.com · …graphs-September-23-2026link only, reading not recorded
- Repositorygithub.com · …bipartite-graphs…link only, reading not recorded
- Lean proofgithub.com · 165.mdpage read word for word
- Lean proofgithub.com · BipartiteCrossing.leanlink only, reading not recorded
- Lean proofgithub.com · CompleteCrossing.leanlink only, reading not recorded
Method
The openai/math repository contains two manuscripts dated September 23, 2026: “The crossing number of complete graphs” and “The crossing number of complete bipartite graphs”. The file lean/docs/165.md describes the scope of the Lean formalisation, with 2 Comparator statements. The family is among the sources of formalization.yaml, and history.md reports no revision. DiscoverIA status: not evaluated, meaning that no third-party evaluation has been found in the sources read. Set on October 12, 2026 from github.com (page read, quotation checked word for word: “describes the scope of the Lean formalization”).
Limits
No named third party: the recorded checks come from the claimant or from a platform tool. The claimant supplies a Lean proof; as of this date, DiscoverIA has found no independent verification of this proof or of the result.
Key facts
- Claimed by
- OpenAI (unreleased internal model)
announced on Oct 6, 2026 - Nature
- new result
- Field
- Mathematics (Combinatorics)
- In openai/math
- family 165, section “Combinatorics”, 2 manuscripts dated Sep 23, 2026
- Lean proof
- supplied by the claimant, marked “unchecked” (not reviewed) in its repository
- Revisions
- none reported by the claimant
- Checked by
- claimant (Lean supplied, not checked by a third party)
- Sources
- 6 : 1 read word for word, 5 links only
- This entry
- AI draft, not reviewed
Status history
Only one observation so far.
- 01
First observation:Not evaluatedCurrent status
Family with a Lean proof supplied by the claimant. No third-party check read; the source marks this proof as not reviewed (“unchecked”).
Source: github.comread word for word
An observation is added; none is ever erased.
In the same set
openai/math: 719 manuscripts in 372 families from an unreleased internal model · entry 159 of 372, by family number
- Not evaluated : Counterexamples to Sidorenko's conjecture and the forcing conjecture (family 161)
- Not evaluated : Counterexamples to Ryser's covering and Gyárfás's tree-cover conjectures (family 162)
- Not evaluated : Hindman's finite sums and products conjecture (family 164)
- Not evaluated : Exact crossing numbers of complete and complete bipartite graphs (family 165)
- Not evaluated : The higher-dimensional Erdős distinct-distances conjecture (family 166)
- Not evaluated : Pinned distances and a power saving for planar unit distances (family 167)
- Not evaluated : Combinatorial invariance of Kazhdan–Lusztig polynomials (family 168)
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