590.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

4 declarations, 4 contain sorry, 0 declared as axioms

Erdős Problem 590

References:

erdosproblems.com/590
[Ch72] Chang, C. C., A partition theorem for the complete graph on {$\omega\sp{\omega }$}. J. Combinatorial Theory Ser. A (1972), 396-452.
[Sp57] Specker, Ernst, Teilmengen von Mengen mit Relationen. Comment. Math. Helv. (1957), 302-314.
[La73] Larson, Jean A., A short proof of a partition theorem for the ordinal {$\omega \sp{\omega }$}. Ann. Math. Logic (1973/74), 129-145.

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/Subsets/FC100SolvedSet1.lean

theorem Erdos590.erdos_590 Let $α$ be the infinite ordinal $\omega^{\omega}$ sorry line 43 source
Source documentation
Let $α$ be the infinite ordinal $\omega^{\omega}$. It was proved by Chang [Ch72] that any red/blue colouring of the edges of $K_α$ there is either a red $K_α$ or a blue $K_3$.

depends on sorry what this means
Statement uses: OrdinalCardinalRamsey, HPow.hPow, Ordinal, Ordinal.omega0, OfNat.ofNat, Cardinal, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
14 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_590 : OrdinalCardinalRamsey (ω ^ ω) (ω ^ ω) 3 := by
…
```
Mentions: OrdinalCardinalRamsey
theorem Erdos590.erdos_590.variants.two Specker [Sp57] proved that when $α=ω^2$ any red/blue colouring of the edges of $K_α$ there is either a red $K_α$ or a blue $K_3$ sorry line 51 source
Source documentation
Specker [Sp57] proved that when $α=ω^2$ any red/blue colouring of the edges of $K_α$ there is either a red $K_α$ or a blue $K_3$.

depends on sorry what this means
Statement uses: OrdinalCardinalRamsey, HPow.hPow, Ordinal, Nat, Ordinal.omega0, OfNat.ofNat, Cardinal
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_590.variants.two : OrdinalCardinalRamsey (ω ^ 2) (ω ^ 2) 3 := by
…
```
Mentions: OrdinalCardinalRamsey
theorem Erdos590.erdos_590.variants.ge_three_false Specker [Sp57] proved that when $α=ω^n$ for $3≤ n < \omega$ then it is not the case that any red/blue colouring of the edges of $K_α$ there is either a red $K_α$ or a blue $K_3$ sorry line 59 source
Source documentation
Specker [Sp57] proved that when $α=ω^n$ for $3≤ n < \omega$ then it is not the case that any red/blue colouring of the edges of $K_α$ there is either a red $K_α$ or a blue $K_3$.

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Not, OrdinalCardinalRamsey, HPow.hPow, Ordinal, Ordinal.omega0, Cardinal
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
20 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_590.variants.ge_three_false {n : ℕ} (h : 3 ≤ n) :
    ¬ OrdinalCardinalRamsey (ω ^ n) (ω ^ n) 3 := by
…
```
Mentions: OrdinalCardinalRamsey
theorem Erdos590.erdos_590.variants.finite_cardinal Let m be a finite cardinal $< \omega$ sorry line 69 source
Source documentation
Let m be a finite cardinal $< \omega$. Let $α$ be the infinite ordinal $\omega^{\omega}$. It was proved by Milnor that any red/blue colouring of the edges of $K_α$ there is either a red $K_α$ or a blue $K_3$. A shorter proof was found by Larson [La73]

depends on sorry what this means
Statement uses: Nat, OrdinalCardinalRamsey, HPow.hPow, Ordinal, Ordinal.omega0, Cardinal
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
9 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_590.variants.finite_cardinal (m : ℕ) : OrdinalCardinalRamsey (ω ^ ω) (ω ^ ω) m := by
…
```
Mentions: OrdinalCardinalRamsey