837.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 837

References:

erdosproblems.com/837

Imports (1)

FormalConjecturesUtil

theorem Erdos837.erdos_837 Documented declaration: erdos_837 sorry line 44 source
Source documentation
Let $k\geq 2$ and $A_k\subseteq [0,1]$ be the set of $\alpha$ such that there exists some $\beta(\alpha)>\alpha$ with the property that, if $G_1,G_2,\ldots$ is a sequence of $k$-uniform hypergraphs with $$\liminf \frac{e(G_n)}{\binom{\lvert G_n\rvert}{k}} >\alpha$$ then there exist subgraphs $H_n\subseteq G_n$ such that $\lvert H_n\rvert \to \infty$ and $$\liminf \frac{e(H_n)}{\binom{\lvert H_n\rvert}{k}} >\beta,$$ and further that this property does not necessarily hold if $>\alpha$ is replaced by $\geq \alpha$. What is $A_3$?

depends on sorry what this means
Statement uses: Eq, Set, Real, Set.ofPred, And, Membership.mem, Set.Icc, OfNat.ofNat, Exists, LT.lt, Hypergraph.HasDensityJump, Nat, Bool.true, Not, Bool.false, sorryAx, Lean.Name, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
16 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_837 :
    {α : ℝ | α ∈ Set.Icc 0 1 ∧ ∃ β : ℝ, α < β ∧
    Hypergraph.HasDensityJump 3 α β true ∧ ¬ Hypergraph.HasDensityJump 3 α β false}
      = answer(sorry) := by
…
```
Mentions: Hypergraph.HasDensityJump