814.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 814

References:

erdosproblems.com/814
[EFRS90] Erdős, P. and Faudree, R. J. and Rousseau, C. C. and Schelp, R. H., Subgraphs of minimal degree k. Discrete Math. (1990), 53--58.
[Er91] Erdős, P., Problems and results in combinatorial analysis and combinatorial number theory. Graph theory, combinatorics, and applications, Vol. 1 (Kalamazoo, MI, 1988) (1991), 397-406.
[Er93] Erdős, Paul, Some of my favorite solved and unsolved problems in graph theory. Quaestiones Math. (1993), 333-350.
[MNS17] Mousset, Frank and Noever, Andreas and Škorić, Nemanja, Smaller subgraphs of minimum degree k. Electron. J. Combin. (2017), Paper No. 4.9, 8.
[Sa19] Sauermann, Lisa, A proof of a conjecture of Erdős, Faudree, Rousseau and Schelp on subgraphs of minimum degree k. J. Combin. Theory Ser. B (2019), 36--75.

Imports (1)

FormalConjecturesUtil

theorem Erdos814.erdos_814 sorry line 59 source
depends on sorry what this means
Statement uses: Iff, True, Nat, GE.ge, OfNat.ofNat, Exists, Real, And, GT.gt, HSub.hSub, SimpleGraph, Fin, Eq, Finset.card, Sym2, SimpleGraph.edgeFinset, Classical.propDecidable, SimpleGraph.Adj, HAdd.hAdd, HMul.hMul and 11 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
58 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos814.erdos_814.variants.at_least sorry line 71 source
depends on sorry what this means
Statement uses: Nat, GE.ge, OfNat.ofNat, Exists, Real, And, GT.gt, HSub.hSub, SimpleGraph, Fin, LE.le, HAdd.hAdd, HMul.hMul, Nat.choose, Finset.card, Sym2, SimpleGraph.edgeFinset, Classical.propDecidable, SimpleGraph.Adj, Finset and 8 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
58 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos814.erdos_814.variants.sauermann Sauermann [Sa19] proved this with $c_k \gg 1/k^3$ sorry line 81 source
Source documentation
Sauermann [Sa19] proved this with $c_k \gg 1/k^3$.

depends on sorry what this means
Statement uses: Exists, Real, And, GT.gt, OfNat.ofNat, Nat, GE.ge, HSub.hSub, SimpleGraph, Fin, LE.le, HAdd.hAdd, HMul.hMul, Nat.choose, Finset.card, Sym2, SimpleGraph.edgeFinset, Classical.propDecidable, SimpleGraph.Adj, Finset and 10 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
72 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_814.variants.sauermann :
    ∃ C > 0, ∀ k ≥ 2, ∀ n ≥ k - 1, ∀ G : SimpleGraph (Fin n),
      (k - 1) * (n + 2 - k) + (k - 2).choose 2 + 1 ≤ G.edgeFinset.card →
        ∃ S : Finset (Fin n), S.Nonempty ∧ (S.card : ℝ) ≤ (1 - C / k ^ 3) * n ∧
          k ≤ (G.induce (S : Set (Fin n))).minDegree := by
…
```