1177.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

3 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 1177

References:

erdosproblems.com/1177

Imports (1)

FormalConjecturesUtil

theorem Erdos1177.erdos_1177.parts.i Let $G$ be a finite $3$-uniform hypergraph, and let $F_G(\kappa)$ denote the collection of $3$-uniform hypergraphs with chromatic number $\kappa$ not containing $G$ sorry line 42 source
Source documentation
Let $G$ be a finite $3$-uniform hypergraph, and let $F_G(\kappa)$ denote the collection of $3$-uniform hypergraphs with chromatic number $\kappa$ not containing $G$. If $F_G(\aleph_1)$ is not empty then there exists $X\in F_G(\aleph_1)$ of cardinality at most $2^{2^{\aleph_0}}$. If both $F_G(\aleph_1)$ and $F_H(\aleph_1)$ are non-empty then $F_G(\aleph_1)\cap F_H(\aleph_1)$ is non-empty. If $\kappa,\lambda$ are uncountable cardinals and $F_G(\kappa)$ is non-empty then $F_G(\lambda)$ is non-empty.

depends on sorry what this means
Statement uses: Fintype, UniformHypergraph, OfNat.ofNat, Nat, UniformHypergraph.HasAvoidingChromaticCardinal, OrderEmbedding, Ordinal, Cardinal, Cardinal.aleph, Exists, DecidableEq, And, Eq, UniformHypergraph.chromaticCardinal, Not, UniformHypergraph.Appears, LE.le, Cardinal.mk, HPow.hPow, Cardinal.aleph0
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
30 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1177.parts.i :
    ∀ (W : Type) [Fintype W] (G : UniformHypergraph W 3),
    G.HasAvoidingChromaticCardinal ℵ₁ →
      ∃ (V : Type) (_ : DecidableEq V) (X : UniformHypergraph V 3),
        X.chromaticCardinal = ℵ₁ ∧ ¬ G.Appears X ∧
          Cardinal.mk V ≤ (2 : Cardinal) ^ ((2 : Cardinal) ^ ℵ₀) := by
…
```
Mentions: UniformHypergraph
theorem Erdos1177.erdos_1177.parts.ii If both $F_G(\aleph_1)$ and $F_H(\aleph_1)$ are non-empty then $F_G(\aleph_1)\cap F_H(\aleph_1)$ is non-empty sorry line 55 source
Source documentation
If both $F_G(\aleph_1)$ and $F_H(\aleph_1)$ are non-empty then $F_G(\aleph_1)\cap F_H(\aleph_1)$ is non-empty.

depends on sorry what this means
Statement uses: Fintype, UniformHypergraph, OfNat.ofNat, Nat, UniformHypergraph.HasAvoidingChromaticCardinal, OrderEmbedding, Ordinal, Cardinal, Cardinal.aleph, Exists, DecidableEq, And, Eq, UniformHypergraph.chromaticCardinal, Not, UniformHypergraph.Appears
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1177.parts.ii :
    ∀ (W U : Type) [Fintype W] [Fintype U]
    (G : UniformHypergraph W 3) (H : UniformHypergraph U 3),
    G.HasAvoidingChromaticCardinal ℵ₁ → H.HasAvoidingChromaticCardinal ℵ₁ →
      ∃ (V : Type) (_ : DecidableEq V) (X : UniformHypergraph V 3),
        X.chromaticCardinal = ℵ₁ ∧ ¬ G.Appears X ∧ ¬ H.Appears X := by
…
```
Mentions: UniformHypergraph
theorem Erdos1177.erdos_1177.parts.iii If $\kappa,\lambda$ are uncountable cardinals and $F_G(\kappa)$ is non-empty then $F_G(\lambda)$ is non-empty sorry line 68 source
Source documentation
If $\kappa,\lambda$ are uncountable cardinals and $F_G(\kappa)$ is non-empty then $F_G(\lambda)$ is non-empty.

depends on sorry what this means
Statement uses: Fintype, UniformHypergraph, OfNat.ofNat, Nat, Cardinal, LT.lt, Cardinal.aleph0, UniformHypergraph.HasAvoidingChromaticCardinal
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
8 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1177.parts.iii :
    ∀ (W : Type) [Fintype W] (G : UniformHypergraph W 3)
    (κ μ : Cardinal), ℵ₀ < κ → ℵ₀ < μ →
      G.HasAvoidingChromaticCardinal κ → G.HasAvoidingChromaticCardinal μ := by
…
```
Mentions: UniformHypergraph