75.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 75

Reference:

[erdosproblems.com/75] (https://www.erdosproblems.com/75)

Imports (1)

FormalConjecturesUtil

theorem Erdos75.erdos_75 Documented declaration: erdos_75 sorry line 39 source
Source documentation
Is there a graph of chromatic number ℵ_ 1 with ℵ_ 1 vertices such that for all ε > 0, if n is sufficiently large and H is a subgraph on n vertices, then H contains an independent set of size > n ^ (1 - ε)?

depends on sorry what this means
Statement uses: Iff, True, Exists, SimpleGraph, And, Eq, Cardinal, SimpleGraph.chromaticCardinal, OrderEmbedding, Ordinal, Cardinal.aleph, OfNat.ofNat, Cardinal.mk, Real, GT.gt, Filter.Eventually, Nat, SimpleGraph.Subgraph, Set.ncard, SimpleGraph.Subgraph.verts and 8 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
45 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_75 :
    answer(sorry) ↔
    ∃ (V : Type) (G : SimpleGraph V),
      G.chromaticCardinal = ℵ_ 1 ∧
      #V = ℵ_ 1 ∧
      ∀ ε > (0 : ℝ),
        ∀ᶠ (n : ℕ) in Filter.atTop, ∀ (H : G.Subgraph),
            H.verts.ncard = n →
            ∃ (I : Finset V),
              (I : Set V) ⊆ H.verts ∧
              G.IsIndepSet (I : Set V) ∧
              (I.card : ℝ) > (n : ℝ) ^ (1 - ε) := by
…
```