490.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

2 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 490

References:

erdosproblems.com/490
[Er61] Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254.
[Er69] Erdős, Paul, Some applications of graph theory to number theory. The Many Facets of Graph Theory (Proc. Conf., Western Mich. Univ., Kalamazoo, Mich., 1968) (1969), 77-82.
[Er72] Erdős, Paul, Extremal problems in number theory. Proceedings of the 1972 Number Theory Conference (Univ. Colorado, Boulder, Colo.) (1972), 80-86.
[Er73] Erdős, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138.
[Sz76] Szemerédi, E., On a problem of P. Erdős. J. Number Theory (1976), 264-270.

Imports (1)

FormalConjecturesUtil

theorem Erdos490.erdos_490 sorry line 61 source
depends on sorry what this means
Statement uses: Iff, True, Exists, Real, Filter.Eventually, Nat, Finset, LE.le, Finset.Icc, OfNat.ofNat, Membership.mem, Eq, HMul.hMul, And, Finset.card, HDiv.hDiv, HPow.hPow, Real.log, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
42 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos490.erdos_490.variants.limit Erdős [Er72] asks whether sorry line 74 source
Source documentation
Erdős [Er72] asks whether $$\lim_{N\to \infty}\max_{A,B\subseteq [N]}\frac{\lvert A\rvert\lvert B\rvert\log N}{N^2}$$ exists, where the maximum is over $A$ and $B$ with all the products $ab$ distinct, and to determine its value.

depends on sorry what this means
Statement uses: Iff, True, Exists, Real, Filter.Tendsto, Nat, HDiv.hDiv, HMul.hMul, SupSet.sSup, Set.ofPred, Finset, And, LE.le, Finset.Icc, OfNat.ofNat, Membership.mem, Eq, Finset.card, Real.log, HPow.hPow and 2 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
48 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_490.variants.limit : answer(sorry) ↔ ∃ L : ℝ, Tendsto (fun N : ℕ =>
    (sSup {x : ℝ | ∃ A B : Finset ℕ, A ⊆ Finset.Icc 1 N ∧ B ⊆ Finset.Icc 1 N ∧
      (∀ a₁ ∈ A, ∀ b₁ ∈ B, ∀ a₂ ∈ A, ∀ b₂ ∈ B, a₁ * b₁ = a₂ * b₂ → a₁ = a₂ ∧ b₁ = b₂) ∧
      x = A.card * B.card}) * Real.log N / N ^ 2) atTop (nhds L) := by
…
```