1037.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

3 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 1037

References:

erdosproblems.com/1037
[Er93] Erdős, Paul, Some of my favorite solved and unsolved problems in graph theory. Quaestiones Math. (1993), 333-350.

Imports (1)

FormalConjecturesUtil

def Erdos1037.IsTrivialSet A set s of vertices of a graph G is trivial if the subgraph it induces is empty or complete, that is, if s is an independent set or a clique of G line 39 source
Source documentation
A set s of vertices of a graph G is trivial if the subgraph it induces is empty or complete, that is, if s is an independent set or a clique of G.

no axioms what this means
Statement uses: SimpleGraph, Set
Proof uses: Or, SimpleGraph.IsIndepSet, SimpleGraph.IsClique
```
def IsTrivialSet {V : Type*} (G : SimpleGraph V) (s : Set V) : Prop :=
…
```
theorem Erdos1037.erdos_1037 Let $G$ be a graph on $n$ vertices in which every degree occurs at most twice, and the number of distinct degrees is $>(\frac{1}{2}+\epsilon)n$ sorry line 54 source
Source documentation
Let $G$ be a graph on $n$ vertices in which every degree occurs at most twice, and the number of distinct degrees is $>(\frac{1}{2}+\epsilon)n$. Must $G$ contain a trivial (empty or complete) subgraph of size 'much larger' than $\log n$?
A question of Chen and Erdős.
The answer is no - Cambie, Chan, and Hunter have in the comment section given a simple construction of a graph on $n$ vertices with at least $\frac{3}{4}n$ distinct degrees, every degree appears at most twice, and the largest trivial subgraph has size $O(\log n)$.

depends on sorry what this means
Statement uses: Iff, False, Real, LT.lt, OfNat.ofNat, Filter.Eventually, Nat, SimpleGraph, Fin, LE.le, Set.ncard, Set.ofPred, Eq, SimpleGraph.neighborSet, HMul.hMul, HAdd.hAdd, HDiv.hDiv, Set.range, Exists, Set and 4 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
40 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1037 : answer(False) ↔
    ∀ ε : ℝ, 0 < ε → ∀ C : ℝ, ∀ᶠ n : ℕ in atTop, ∀ G : SimpleGraph (Fin n),
      (∀ d : ℕ, {v : Fin n | (G.neighborSet v).ncard = d}.ncard ≤ 2) →
      ((1 / 2 + ε) * (n : ℝ) <
        ((Set.range fun v : Fin n => (G.neighborSet v).ncard).ncard : ℝ)) →
      ∃ s : Set (Fin n), IsTrivialSet G s ∧ (C * Real.log (n : ℝ) < (s.ncard : ℝ)) := by
…
```
Mentions: Erdos1037.IsTrivialSet
theorem Erdos1037.erdos_1037.variants.cambie_chan_hunter Documented declaration: cambie_chan_hunter sorry line 68 source
Source documentation
Cambie, Chan, and Hunter have in the comment section given a simple construction of a graph on $n$ vertices with at least $\frac{3}{4}n$ distinct degrees, every degree appears at most twice, and the largest trivial subgraph has size $O(\log n)$.

depends on sorry what this means
Statement uses: Exists, Real, Filter.Eventually, Nat, SimpleGraph, Fin, And, LE.le, Set.ncard, Set.ofPred, Eq, SimpleGraph.neighborSet, OfNat.ofNat, HMul.hMul, HDiv.hDiv, Set.range, Set, Erdos1037.IsTrivialSet, Real.log, Filter.atTop
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_1037.variants.cambie_chan_hunter :
    ∃ C : ℝ, ∀ᶠ n : ℕ in atTop, ∃ G : SimpleGraph (Fin n),
      (∀ d : ℕ, {v : Fin n | (G.neighborSet v).ncard = d}.ncard ≤ 2) ∧
      (3 / 4 * (n : ℝ) ≤
        ((Set.range fun v : Fin n => (G.neighborSet v).ncard).ncard : ℝ)) ∧
      ∀ s : Set (Fin n), IsTrivialSet G s → ((s.ncard : ℝ) ≤ C * Real.log (n : ℝ)) := by
…
```
Mentions: Erdos1037.IsTrivialSet