765.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 765

References:

erdosproblems.com/765
[Er38] Erdős, P., On sequences of integers no one of which divides the product of two others and on related problems. Tomsk. Gos. Univ. Ucen Zap. (1938), 74-82.
[Re58] Reiman, I., Über ein Problem von K. Zarankiewicz. Acta Math. Acad. Sci. Hungar. 9 (1958), 269-273.
[ERS66] Erdős, P. and Rényi, A. and Sós, V. T., On a problem of graph theory. Studia Sci. Math. Hungar. (1966), 215--235.
[Br66] Brown, W. G., On graphs that do not contain a Thomsen graph. Canad. Math. Bull. (1966), 281-285.
[Er75] Erdős, P., Some recent progress on extremal problems in graph theory. Congr. Numer. (1975), 3-14.
[Fu83] Füredi, Z., Graphs without quadrilaterals. J. Combin. Theory Ser. B 34 (1983), 187-190.
[Er93] Erdős, Paul, Some of my favorite solved and unsolved problems in graph theory. Quaestiones Math. (1993), 333-350.
[MaYa23] Ma, Jie and Yang, Tianchi, Upper bounds on the extremal number of the 4-cycle. Bull. Lond. Math. Soc. (2023), 1655--1667.

Imports (1)

FormalConjecturesUtil

theorem Erdos765.erdos_765 sorry line 62 source
depends on sorry what this means
Statement uses: Asymptotics.IsEquivalent, Nat, Real, Filter.atTop, SimpleGraph.extremalNumber, Fin, OfNat.ofNat, SimpleGraph.cycleGraph, HDiv.hDiv, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
34 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos765.erdos_765.variants.second_term Documented declaration: second_term sorry line 74 source
Source documentation
Erdős [Er93] conjectured that $\mathrm{ex}(n; C_4) = \frac{n^{3/2}}{2} + \frac n4 + O(n^{1/2})$ for all $n$, having proved the upper bound $\mathrm{ex}(n; C_4) \le \frac{n^{3/2}}{2} + \frac n4 + O(n^{1/2})$ in [Er75]. This is false: Ma and Yang [MaYa23] proved that, for some absolute constant $c > 0$ and a positive density set of $n$, $\mathrm{ex}(n; C_4) \le \frac{n^{3/2}}{2} + (\frac14 - c) n$.

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

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_765.variants.second_term : answer(False) ↔
    (fun n : ℕ ↦ (SimpleGraph.extremalNumber n (SimpleGraph.cycleGraph 4) : ℝ) -
      (n : ℝ) ^ (3 / 2 : ℝ) / 2 - (n : ℝ) / 4) =O[atTop] fun n : ℕ ↦ (n : ℝ) ^ (1 / 2 : ℝ) := by
…
```