1014.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, 2 contain sorry, 0 declared as axioms

Erdős Problem 1014

References:

erdosproblems.com/1014
[Er71] Erdős, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) (1971), 97-109.
[OpenAI26] On the ratio of $R(k,\ell)$ and $R(k,\ell+1)$, proof due to an internal model at OpenAI (2026). https://cdn.openai.com/pdf/6dc7175d-d9e7-4b8d-96b8-48fe5798cd5b/Ramsey.pdf

Imports (1)

FormalConjecturesUtil

theorem Erdos1014.erdos_1014 Let $R(k,l)$ be the Ramsey number, so the minimal $n$ such that every graph on at least $n$ vertices contains either a $K_k$ or an independent set on $l$ vertices sorry line 54 source
Source documentation
Let $R(k,l)$ be the Ramsey number, so the minimal $n$ such that every graph on at least $n$ vertices contains either a $K_k$ or an independent set on $l$ vertices.
Prove, for fixed $k\geq 3$, that $$\lim_{l\to \infty}\frac{R(k,l+1)}{R(k,l)}=1.$$
This has been solved by an internal model at OpenAI.

L4DEL explanation
Let $R(k,l)$ be the Ramsey number, so the minimal $n$ such that every graph on at least $n$ vertices contains either a $K_k$ or an independent set on $l$ vertices. Prove, for fixed $k\geq 3$, that $$\lim_{l\to \infty}\frac{R(k,l+1)}{R(k,l)}=1.$$ This has been solved by an internal model at OpenAI.
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Filter.Tendsto, Real, HDiv.hDiv, SimpleGraph.classicalRamsey, HAdd.hAdd, Filter.atTop, nhds
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_1014 : ∀ k : ℕ, 3 ≤ k →
    Tendsto (fun l : ℕ ↦ (R(k, l + 1) : ℝ) / (R(k, l) : ℝ)) atTop (𝓝 1) := by
…
```
theorem Erdos1014.erdos_1014.variants.upper_bound That proof in fact shows that $$R(k,l+1)\leq (1+O(l^{-c/k^2}))R(k,l)$$ for some constant $c>0$ sorry line 64 source
Source documentation
That proof in fact shows that $$R(k,l+1)\leq (1+O(l^{-c/k^2}))R(k,l)$$ for some constant $c>0$.

depends on sorry what this means
Statement uses: Exists, Real, And, LT.lt, OfNat.ofNat, Nat, LE.le, Filter.Eventually, SimpleGraph.classicalRamsey, HAdd.hAdd, HMul.hMul, HPow.hPow, HDiv.hDiv, Neg.neg, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
50 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1014.variants.upper_bound :
    ∃ c : ℝ, 0 < c ∧ ∀ k : ℕ, 3 ≤ k → ∃ C : ℝ, ∀ᶠ l : ℕ in atTop,
      (R(k, l + 1) : ℝ) ≤ (1 + C * (l : ℝ) ^ (-c / (k : ℝ) ^ 2)) * (R(k, l) : ℝ) := by
…
```
Mentions: c