146.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 10 curated AI-written annotations and 7190 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

2 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 146

References:

erdosproblems.com/146
[ErSi84] Erdős, P. and Simonovits, M., Cube-supersaturated graphs and related problems. Progress in graph theory (1984), 203-218.
[OpenAI26] OpenAI, Ten advances in mathematics and theoretical computer science. (2026).

Imports (1)

FormalConjecturesUtil

theorem Erdos146.erdos_146 If $H$ is bipartite and is $r$-degenerate, that is, every induced subgraph of $H$ has minimum degree $\leq r$, then $$\mathrm{ex}(n;H) \ll n^{2-1/r}.$$ The answer is no sorry line 46 source
Source documentation
If $H$ is bipartite and is $r$-degenerate, that is, every induced subgraph of $H$ has minimum degree $\leq r$, then $$\mathrm{ex}(n;H) \ll n^{2-1/r}.$$
The answer is no. OpenAI [OpenAI26] give a connected bipartite 2-degenerate H and constants c, ε > 0 with $\mathrm{ex}(n;H)\geq cn^{3/2+\epsilon}$ for all large n, which exceeds the conjectured $n^{2-1/2}=n^{3/2}$. See erdos_146.variants.two_degenerate_counterexample.

depends on sorry what this means
Statement uses: Iff, False, Nat, SimpleGraph, Fin, LT.lt, OfNat.ofNat, SimpleGraph.IsBipartite, SimpleGraph.IsDegenerate, Asymptotics.IsBigO, Real, Filter.atTop, SimpleGraph.extremalNumber, HPow.hPow, HSub.hSub, HDiv.hDiv
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
36 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_146 : answer(False) ↔
    ∀ (r q : ℕ) (H : SimpleGraph (Fin q)),
      0 < r → H.IsBipartite → H.IsDegenerate r →
        Asymptotics.IsBigO atTop
          (fun n : ℕ => (extremalNumber n H : ℝ))
          (fun n : ℕ => (n : ℝ) ^ ((2 : ℝ) - 1 / (r : ℝ))) := by
…
```
theorem Erdos146.erdos_146.variants.two_degenerate_counterexample sorry line 60 source
depends on sorry what this means
Statement uses: Exists, Nat, SimpleGraph, Fin, And, SimpleGraph.Connected, SimpleGraph.IsBipartite, SimpleGraph.IsDegenerate, OfNat.ofNat, Real, LT.lt, Filter.Eventually, LE.le, HMul.hMul, HPow.hPow, HAdd.hAdd, HDiv.hDiv, SimpleGraph.extremalNumber, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
40 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.