1009.lean`

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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

2 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 1009

References:

erdosproblems.com/1009
[Er71] Erdős, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) (1971), 97--109.
[Gy88] Győri, E., On the number of edge-disjoint triangles in graphs of given size. (1988), 267--276.

Imports (1)

FormalConjecturesUtil

theorem Erdos1009.erdos_1009 sorry line 54 source
depends on sorry what this means
Statement uses: Iff, True, Real, LT.lt, OfNat.ofNat, Exists, Nat, SimpleGraph, Fin, LE.le, HAdd.hAdd, HDiv.hDiv, HPow.hPow, Set.ncard, Sym2, SimpleGraph.edgeSet, HMul.hMul, Finset, And, Membership.mem and 4 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 Erdos1009.erdos_1009.variants.sauer Documented declaration: sauer sorry line 66 source
Source documentation
Sauer gave an example of a graph on $n$ vertices with $\lfloor n^2/4\rfloor+2n-6$ many edges which contains only $2n-7$ many edge disjoint triangles: if $n=2r+4$ then $G$ is the complete tripartite graph on $[r]\times [r]\times [4]$, with a $K_4$ on the $[4]$ vertices also.

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Exists, SimpleGraph, Fin, HAdd.hAdd, HMul.hMul, And, Eq, Set.ncard, Sym2, SimpleGraph.edgeSet, HSub.hSub, HDiv.hDiv, HPow.hPow, Finset, Membership.mem, SimpleGraph.IsNClique, Set.Pairwise and 2 more
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 erdos_1009.variants.sauer : ∀ r : ℕ, 1 ≤ r → ∃ G : SimpleGraph (Fin (2 * r + 4)),
    G.edgeSet.ncard = (2 * r + 4) ^ 2 / 4 + 2 * (2 * r + 4) - 6 ∧
      ∀ T : Finset (Finset (Fin (2 * r + 4))), (∀ t ∈ T, G.IsNClique 3 t) →
        (T : Set (Finset (Fin (2 * r + 4)))).Pairwise (fun s t => (s ∩ t).card ≤ 1) →
          T.card ≤ 2 * (2 * r + 4) - 7 := by
…
```