133.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

7 declarations, 6 contain sorry, 0 declared as axioms

Erdős Problem 133

References:

erdosproblems.com/133
[Er97b] Erdős, Paul, Some old and new problems in various branches of combinatorics. Discrete Math. (1997), 227-231.
[HaSe84] Hanson, D. and Seyffarth, K., $k$-saturated graphs of prescribed maximum degree. Congr. Numer. (1984), 169-182.
[FuSe94] Füredi, Zoltán and Seress, Ákos, Maximal triangle-free graphs with restrictions on the degrees. J. Graph Theory (1994), 11-24.
[HaLe18] Haviv, Ishay and Levy, Dan, Symmetric complete sum-free sets in cyclic groups. Israel J. Math. (2018), 931-956.

Imports (1)

FormalConjecturesUtil

def Erdos133.f f n is the least possible maximum degree of a triangle-free graph on n vertices with diameter 2, i.e. the largest f such that every such graph has a vertex of degree ≥ f line 46 source
Source documentation
f n is the least possible maximum degree of a triangle-free graph on n vertices with diameter 2, i.e. the largest f such that every such graph has a vertex of degree ≥ f.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: InfSet.sInf, Set.ofPred, Exists, SimpleGraph, Fin, And, SimpleGraph.CliqueFree, OfNat.ofNat, Eq, SimpleGraph.diam, LE.le, SimpleGraph.degree, Membership.mem, Set, SimpleGraph.neighborSet, Classical.propDecidable, SimpleGraph.Adj
7 instances, coercions and auxiliary constants left out.
```
noncomputable def f (n : ℕ) : ℕ :=
…
```
theorem Erdos133.erdos_133 sorry line 67 source
depends on sorry what this means
Statement uses: Iff, False, Filter.Tendsto, Nat, Real, HDiv.hDiv, Erdos133.f, Real.sqrt, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
15 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos133.erdos_133.variants.isTheta sorry line 73 source
depends on sorry what this means
Statement uses: Asymptotics.IsTheta, Nat, Real, Filter.atTop, Erdos133.f, Real.sqrt
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
9 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos133.erdos_133.variants.lower_bound The lower bound $f(n)\geq (1-o(1))\sqrt{n}$ follows from the fact that a graph with maximum degree $d$ and diameter $2$ has at most $d^2+1$ many vertices sorry line 81 source
Source documentation
The lower bound $f(n)\geq (1-o(1))\sqrt{n}$ follows from the fact that a graph with maximum degree $d$ and diameter $2$ has at most $d^2+1$ many vertices.

depends on sorry what this means
Statement uses: Real, LT.lt, OfNat.ofNat, Filter.Eventually, Nat, LE.le, HMul.hMul, HSub.hSub, Real.sqrt, Erdos133.f, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
27 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_133.variants.lower_bound : ∀ ε : ℝ, 0 < ε →
    ∀ᶠ n : ℕ in atTop, (1 - ε) * √n ≤ f n := by
…
```
Mentions: Erdos133.f
theorem Erdos133.erdos_133.variants.hanson_seyffarth Hanson and Seyffarth [HaSe84] proved that $f(n)\leq (\sqrt{2}+o(1))\sqrt{n}$ sorry line 87 source
Source documentation
Hanson and Seyffarth [HaSe84] proved that $f(n)\leq (\sqrt{2}+o(1))\sqrt{n}$.

depends on sorry what this means
Statement uses: Real, LT.lt, OfNat.ofNat, Filter.Eventually, Nat, LE.le, Erdos133.f, HMul.hMul, HAdd.hAdd, Real.sqrt, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
30 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_133.variants.hanson_seyffarth : ∀ ε : ℝ, 0 < ε →
    ∀ᶠ n : ℕ in atTop, (f n : ℝ) ≤ (√2 + ε) * √n := by
…
```
Mentions: Erdos133.f
theorem Erdos133.erdos_133.variants.furedi_seress Füredi and Seress [FuSe94] proved that $f(n)\leq (\frac{2}{\sqrt{3}}+o(1))\sqrt{n}$ sorry line 93 source
Source documentation
Füredi and Seress [FuSe94] proved that $f(n)\leq (\frac{2}{\sqrt{3}}+o(1))\sqrt{n}$.

depends on sorry what this means
Statement uses: Real, LT.lt, OfNat.ofNat, Filter.Eventually, Nat, LE.le, Erdos133.f, HMul.hMul, HAdd.hAdd, HDiv.hDiv, Real.sqrt, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
36 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_133.variants.furedi_seress : ∀ ε : ℝ, 0 < ε →
    ∀ᶠ n : ℕ in atTop, (f n : ℝ) ≤ (2 / √3 + ε) * √n := by
…
```
Mentions: Erdos133.f
theorem Erdos133.erdos_133.variants.asymptotic Is $f(n)\sim \sqrt{n}$? sorry line 99 source
Source documentation
Is $f(n)\sim \sqrt{n}$? Alon believes that this is the truth.

depends on sorry what this means
Statement uses: Iff, True, Asymptotics.IsEquivalent, Nat, Real, Filter.atTop, Erdos133.f, Real.sqrt
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
17 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_133.variants.asymptotic :
    answer(sorry) ↔ (fun n : ℕ ↦ (f n : ℝ)) ~[atTop] fun n ↦ √n := by
…
```
Mentions: Erdos133.f