1219.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, 1 contain sorry, 0 declared as axioms

Erdős Problem 1219

References:

erdosproblems.com/1219
[ErHa71] Erdős, Paul and Hajnal, András, Unsolved problems in set theory. Axiomatic Set Theory, Proc. Sympos. Pure Math. XIII Part I (1971), 17-48.
[Ko25b] Komjáth, Péter, The Erdős–Hajnal problem list. Bull. Symb. Log. (2025), 418-461.
[Sh75] Shelah, Saharon, Notes on partition calculus. Infinite and finite sets (Colloq., Keszthely, 1973), Colloq. Math. Soc. János Bolyai 10, North-Holland (1975), 1257-1276.

Imports (1)

FormalConjecturesUtil

theorem Erdos1219.erdos_1219 sorry line 53 source
depends on sorry what this means
Statement uses: Iff, True, Nat, StrictMono, Cardinal, HPow.hPow, OfNat.ofNat, OrderEmbedding, Ordinal, Cardinal.aleph, LT.lt, Ordinal.omega0, Combinatorics.cardinalPartitionRel, Cardinal.sum, Ordinal.ToType
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
38 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos1219.erdos_1219.test.aleph_omega_lt_sum Under the hypotheses of erdos_1219, the target $\aleph_\omega$ is smaller than the source $\sum_k 2^{\aleph_{n_k}}$, so the partition relation is not ruled out by cardinality line 65 source
Source documentation
Under the hypotheses of erdos_1219, the target $\aleph_\omega$ is smaller than the source $\sum_k 2^{\aleph_{n_k}}$, so the partition relation is not ruled out by cardinality.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LT.lt, Cardinal, OrderEmbedding, Ordinal, Cardinal.aleph, Ordinal.omega0, HPow.hPow, OfNat.ofNat, Cardinal.sum
Proof uses: LT.lt.trans_le, Cardinal.le_sum
36 instances, coercions and auxiliary constants left out.
```
theorem erdos_1219.test.aleph_omega_lt_sum (n : ℕ → ℕ)
    (h0 : ℵ_ ω < (2 : Cardinal.{u}) ^ ℵ_ (n 0)) :
    ℵ_ ω < sum fun k => (2 : Cardinal.{u}) ^ ℵ_ (n k) :=
…
```