714.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

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

1 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 714

References:

erdosproblems.com/714
[Br66] Brown, W. G., On graphs that do not contain a Thomsen graph. Canad. Math. Bull. (1966), 281-285.
[ERS66] Erdős, P. and Rényi, A. and Sós, V. T., On a problem of graph theory. Studia Sci. Math. Hungar. (1966), 215--235.
[KST54] Kövari, T. and Sós, V. T. and Turán, P., On a problem of K. Zarankiewicz. Colloq. Math. (1954), 50-57.

Imports (1)

FormalConjecturesUtil

theorem Erdos714.erdos_714 Is it true that $$\mathrm{ex}(n; K_{r,r}) \gg n^{2-1/r}?$$ sorry line 43 source
Source documentation
Is it true that $$\mathrm{ex}(n; K_{r,r}) \gg n^{2-1/r}?$$

depends on sorry what this means
Statement uses: Iff, True, Nat, LE.le, OfNat.ofNat, Exists, Real, And, LT.lt, Filter.Eventually, HMul.hMul, HPow.hPow, HSub.hSub, HDiv.hDiv, SimpleGraph.extremalNumber, Sum, Fin, completeBipartiteGraph, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
46 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_714 : answer(sorry) ↔
    ∀ r : ℕ, 2 ≤ r → ∃ c : ℝ, 0 < c ∧ ∀ᶠ n : ℕ in atTop,
      c * (n : ℝ) ^ ((2 : ℝ) - 1 / (r : ℝ)) ≤
        (extremalNumber n (completeBipartiteGraph (Fin r) (Fin r)) : ℝ) := by
…
```
Mentions: c