803.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 803

References:

erdosproblems.com/803
[ErSi70] Erdős, P. and Simonovits, M., Some extremal problems in graph theory. Combinatorial theory and its applications, I-III (Proc. Colloq., Balatonfüred, 1969) (1970), 377-390.
[Al08] Alon, Noga, Problems and results in extremal combinatorics. II. Discrete Math. (2008), 4460-4472.
[JaSu23] Janzer, Oliver and Sudakov, Benny, Resolution of the Erdős-Sauer problem on regular subgraphs. Forum Math. Pi (2023), Paper No. e19, 13.

Imports (1)

FormalConjecturesUtil

theorem Erdos803.erdos_803 sorry line 56 source
depends on sorry what this means
Statement uses: Iff, False, Exists, Real, And, LT.lt, OfNat.ofNat, Nat, GE.ge, Filter.Eventually, SimpleGraph, Fin, LE.le, HMul.hMul, Real.log, Set.ncard, Sym2, SimpleGraph.edgeSet, SimpleGraph.Subgraph, Eq and 9 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
32 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos803.erdos_803.variants.alon Documented declaration: alon sorry line 68 source
Source documentation
Alon [Al08] proved that for every $D>1$ and large $n$ there is a graph $G$ with $n$ vertices and $\geq n\log n$ edges such that if $H$ is a $D$-balanced subgraph on $m$ vertices then $H$ has $\ll m\sqrt{\log m}+\log D$ many edges.

depends on sorry what this means
Statement uses: Exists, Real, LT.lt, OfNat.ofNat, Filter.Eventually, Nat, SimpleGraph, Fin, And, LE.le, HMul.hMul, Real.log, Set.ncard, Sym2, SimpleGraph.edgeSet, SimpleGraph.Subgraph, SimpleGraph.IsBalanced, SimpleGraph.Subgraph.verts, Membership.mem, Set and 7 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
33 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_803.variants.alon :
    ∃ C : ℝ, ∀ D : ℝ, 1 < D → ∀ᶠ n : ℕ in atTop, ∃ G : SimpleGraph (Fin n),
      (n : ℝ) * log n ≤ G.edgeSet.ncard ∧
      ∀ H : G.Subgraph, IsBalanced H.coe D →
        (H.edgeSet.ncard : ℝ) ≤ C * (H.verts.ncard * √(log H.verts.ncard) + log D) := by
…
```
theorem Erdos803.erdos_803.variants.janzer_sudakov Documented declaration: janzer_sudakov sorry line 80 source
Source documentation
Janzer and Sudakov [JaSu23] have proved that, for any $k$, if $n$ is sufficiently large then any graph on $n$ vertices with at least $n\log n$ edges contains a $O(1)$-balanced subgraph on $m\geq k$ vertices with $\gg_k \frac{\sqrt{\log m}}{(\log\log m)^{3/2}}m$ many edges.

depends on sorry what this means
Statement uses: Exists, Real, Nat, And, GT.gt, OfNat.ofNat, Filter.Eventually, SimpleGraph, Fin, LE.le, HMul.hMul, Real.log, Set.ncard, Sym2, SimpleGraph.edgeSet, SimpleGraph.Subgraph, SimpleGraph.Subgraph.verts, SimpleGraph.IsBalanced, Membership.mem, Set and 8 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
48 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_803.variants.janzer_sudakov :
    ∃ D : ℝ, ∀ k : ℕ, ∃ c > 0, ∀ᶠ n : ℕ in atTop, ∀ G : SimpleGraph (Fin n),
      (n : ℝ) * log n ≤ G.edgeSet.ncard →
        ∃ H : G.Subgraph, k ≤ H.verts.ncard ∧ IsBalanced H.coe D ∧
          c * (√(log H.verts.ncard) / (log (log H.verts.ncard)) ^ (3 / 2 : ℝ)) *
            H.verts.ncard ≤ H.edgeSet.ncard := by
…
```
Mentions: c