1010.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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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 1010

References:

erdosproblems.com/1010
[Er62d] Erdős, P., On a theorem of Rademacher-Turán. Illinois J. Math. (1962), 122-127.
[LoSi76] Lovász, L. and Simonovits, Miklós, On the number of complete subgraphs of a graph. (1976), 431-441.
[NiKh81] Nikiforov, V. S. and Khadzhiivanov, N. G., Solution of the problem of P. Erdős on the number of triangles in graphs with $n$ vertices and $[n^2/4]+l$ edges. C. R. Acad. Bulgare Sci. (1981), 969-970.

Imports (1)

FormalConjecturesUtil

theorem Erdos1010.erdos_1010 sorry line 54 source
depends on sorry what this means
Statement uses: Iff, True, Nat, LT.lt, HDiv.hDiv, OfNat.ofNat, SimpleGraph, Fin, Eq, Finset.card, Sym2, SimpleGraph.edgeFinset, Classical.propDecidable, SimpleGraph.Adj, HAdd.hAdd, HPow.hPow, LE.le, HMul.hMul, Finset, SimpleGraph.cliqueFinset
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
34 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos1010.erdos_1010.variants.rademacher Rademacher proved that every graph on $n$ vertices with $\lfloor n^2/4\rfloor+1$ edges contains at least $\lfloor n/2\rfloor$ triangles sorry line 64 source
Source documentation
Rademacher proved that every graph on $n$ vertices with $\lfloor n^2/4\rfloor+1$ edges contains at least $\lfloor n/2\rfloor$ triangles.

depends on sorry what this means
Statement uses: Nat, SimpleGraph, Fin, Eq, Finset.card, Sym2, SimpleGraph.edgeFinset, Classical.propDecidable, SimpleGraph.Adj, HAdd.hAdd, HDiv.hDiv, HPow.hPow, OfNat.ofNat, LE.le, Finset, SimpleGraph.cliqueFinset
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1010.variants.rademacher : ∀ n : ℕ, ∀ G : SimpleGraph (Fin n),
    G.edgeFinset.card = n ^ 2 / 4 + 1 → n / 2 ≤ (G.cliqueFinset 3).card := by
…
```
theorem Erdos1010.erdos_1010.variants.erdos Erdős [Er62d] proved that every graph on $n$ vertices with $\lfloor n^2/4\rfloor+t$ edges contains at least $t\lfloor n/2\rfloor$ triangles, for all $t0$ sorry line 74 source
Source documentation
Erdős [Er62d] proved that every graph on $n$ vertices with $\lfloor n^2/4\rfloor+t$ edges contains at least $t\lfloor n/2\rfloor$ triangles, for all $t<cn$, for some constant $c>0$.

L4DEL explanation
Erdős [Er62d] proved that every graph on $n$ vertices with $\lfloor n^2/4\rfloor+t$ edges contains at least $t\lfloor n/2\rfloor$ triangles, for all $t0$.
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Exists, Real, And, LT.lt, OfNat.ofNat, Nat, HMul.hMul, SimpleGraph, Fin, Eq, Finset.card, Sym2, SimpleGraph.edgeFinset, Classical.propDecidable, SimpleGraph.Adj, HAdd.hAdd, HDiv.hDiv, HPow.hPow, LE.le, Finset and 1 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
44 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1010.variants.erdos : ∃ c : ℝ, 0 < c ∧ ∀ n t : ℕ, t < c * n →
    ∀ G : SimpleGraph (Fin n), G.edgeFinset.card = n ^ 2 / 4 + t →
      t * (n / 2) ≤ (G.cliqueFinset 3).card := by
…
```
Mentions: c