915.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 915

References:

erdosproblems.com/915
[BoEr62] Bollobás, Béla and Erdős, Pál, Extremal problems in graph theory. Mat. Lapok (1962), 143--152.
[Er67b] Erdős, Paul, Extremal problems in graph theory. A Seminar on Graph Theory (1967), 54-59.
[Ba60] P. Bártfai, Solution of a problem posed by P. Erdős. Mat. Lapok (1960), 175-176.
[Bo66] Bollobás, B., On graphs with at most three independent paths connecting any two vertices. Studia Sci. Math. Hungar. (1966), 137--140.
[Le73] Leonard, John L., On a conjecture of Bollobás and Erdős. Period. Math. Hungar. (1973), 281--284.
[Ma73] Mader, W., Ein Extremalproblem des Zusammenhangs von Graphen. Math. Z. (1973), 223--231.
[SoTh74] Sørensen, Bo Aagaard and Thomassen, Carsten, On $k$-rails in graphs. J. Combinatorial Theory Ser. B (1974), 143--159.

Imports (1)

FormalConjecturesUtil

theorem Erdos915.erdos_915 sorry line 64 source
depends on sorry what this means
Statement uses: Iff, False, Nat, LE.le, OfNat.ofNat, Fintype, SimpleGraph, Eq, Fintype.card, HAdd.hAdd, HMul.hMul, HSub.hSub, Set.ncard, Sym2, SimpleGraph.edgeSet, Nat.choose, Exists, And, Ne, Finset and 6 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
22 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos915.erdos_915.variants.edge_disjoint The edge-disjoint reading of erdős_915 sorry line 80 source
Source documentation
The edge-disjoint reading of erdős_915. Mader [Ma73] proved that if a graph with $n$ vertices has more than $$\frac{m}{2}(n-1)-\frac{1}{2}(e_0(G)+\cdots+e_{m-2}(G))$$ edges then $G$ contains two vertices connected by $m$ edge-disjoint paths (where $e_r(G)$ counts the number of vertices of degree $\leq r$). In particular, this confirms (and is stronger than) the conjecture.

L4DEL explanation
The edge-disjoint reading of erdős_915. Mader [Ma73] proved that if a graph with $n$ vertices has more than $$\frac{m}{2}(n-1)-\frac{1}{2}(e_0(G)+\cdots+e_{m-2}(G))$$ edges then $G$ contains two vertices connected by $m$ edge-disjoint paths (where $e_r(G)$ counts the number of vertices of degree $\leq r$). In particular, this confirms (and is stronger than) the conjecture.
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Iff, True, Nat, LE.le, OfNat.ofNat, Fintype, SimpleGraph, Eq, Fintype.card, HAdd.hAdd, HMul.hMul, HSub.hSub, Set.ncard, Sym2, SimpleGraph.edgeSet, Nat.choose, Exists, And, Ne, Finset and 7 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
22 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_915.variants.edge_disjoint : answer(True) ↔
    ∀ m n : ℕ, 2 ≤ m → 1 ≤ n → ∀ (V : Type) [Fintype V] (G : SimpleGraph V),
      Fintype.card V = 1 + n * (m - 1) → G.edgeSet.ncard = 1 + n * m.choose 2 →
        ∃ u v : V, u ≠ v ∧ ∃ P : Finset (G.Walk u v), P.card = m ∧ (∀ p ∈ P, p.IsPath) ∧
          (P : Set (G.Walk u v)).Pairwise fun p q => List.Disjoint p.edges q.edges := by
…
```