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Ten Advances in Mathematics and Theoretical Computer Science

OpenAI

A connected bipartite two-degenerate graph H meeting the stated degree conditions has extremal number at least c n^(3/2+ε) for some c,ε>0. This contradicts the proposed exponent for that class.

Sources for this record

What was already known

Erdős #146 proposed an n^(3/2)-scale upper growth rate for the specified two-degenerate bipartite class.

AI involvement AI-led

OpenAI attributes the construction and Lean formalization to Astra. Humans selected the problem and prepared the manuscript.

OpenAI’s original disclosure assigns the argument and subsequent Lean formalization to Astra; human manuscript preparation is separately credited.

Importance Notable

Resolves the named extremal-graph question by counterexample.

Limitations

The result concerns a graph meeting the chapter’s degree conditions. Human reviewers requested an expanded optimization in Lemma 5.1; the Lean certificate is source-reported.

Technical verification Formal (Lean 4.32.0 with Mathlib and Comparator + human)

The project reports a complete Lean certificate for the stated formal theorem. Human assessments have narrower scope; semantic matching and any manuscript gaps remain separately noted.

Independent checking

chapter-specific mathematical review of the published argument, with the limited coverage summarized in the audit. Table 1; Chapter 10 discussion

Checker
Chapter 10 human reviewers in the Sienicki et al. review corpus
Scope
both main Chapter 10 results, with a missing displayed optimization in Lemma 5.1
Basis
Reported by the linked checker
Evidence source

Formal checking

Lean 4.32.0 formal certificate in CompactnessAndDegeneracy.lean; comparator target for TwoDegenerateGraphs.twoDegenerateExtremalCounterexample; manifest reports zero sorry and only standard axioms. main_results entry TwoDegenerateGraphs.twoDegenerateExtremalCounterexample; status and axioms

Checker
OpenAI ten-proofs project, Lean kernel and Comparator configuration
Scope
two-degenerate bipartite extremal counterexample with exponent above 3/2
Basis
Reported by the linked checker
Evidence source

Checking a formal statement does not establish that it matches the original problem.

Human understanding Full

The two Chapter 10 assessments and manuscript explain the layered/sampled graph construction and forbidden-subgraph argument; the remaining requested optimization is an expository calculation.

Correction and publication history