755.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

10 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 755

References:

erdosproblems.com/755
[ErPu75] Erdős, Paul and Purdy, George, Some extremal problems in geometry. III. (1975), 291--308.
[Er94b] Erdős, Paul, Some problems in number theory, combinatorics and combinatorial geometry. Math. Pannon. (1994), 261--269.
[CDL25b] Clemen, Felix Christian, Dumitrescu, Adrian, and Liu, Dingyuan, The number of regular simplices in higher dimensions. arXiv:2507.19841 (2025).

Imports (1)

FormalConjecturesUtil

def Erdos755.IsEquilateralTriangle A three-point set whose pairwise distances are all equal to side line 41 source
Source documentation
A three-point set whose pairwise distances are all equal to side.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real, Finset, EuclideanSpace, Fin
Proof uses: And, Eq, Finset.card, OfNat.ofNat, Membership.mem, Ne, Dist.dist, ENNReal
12 instances, coercions and auxiliary constants left out.
```
def IsEquilateralTriangle {d : ℕ} (side : ℝ)
    (T : Finset (EuclideanSpace ℝ (Fin d))) : Prop :=
…
```
def Erdos755.IsUnitEquilateralTriangle A unit equilateral triangle in Euclidean d-space line 46 source
Source documentation
A unit equilateral triangle in Euclidean d-space.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Finset, EuclideanSpace, Real, Fin
Proof uses: Erdos755.IsEquilateralTriangle, OfNat.ofNat
2 instances, coercions and auxiliary constants left out.
```
def IsUnitEquilateralTriangle {d : ℕ}
    (T : Finset (EuclideanSpace ℝ (Fin d))) : Prop :=
…
```
def Erdos755.IsAnySizeEquilateralTriangle An equilateral triangle of any positive side length in Euclidean d-space line 51 source
Source documentation
An equilateral triangle of any positive side length in Euclidean d-space.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Finset, EuclideanSpace, Real, Fin
Proof uses: Exists, And, LT.lt, OfNat.ofNat, Erdos755.IsEquilateralTriangle
3 instances, coercions and auxiliary constants left out.
```
def IsAnySizeEquilateralTriangle {d : ℕ}
    (T : Finset (EuclideanSpace ℝ (Fin d))) : Prop :=
…
```
def Erdos755.unitEquilateralTriangleCount Number of unit equilateral triangles spanned by a finite point set line 56 source
Source documentation
Number of unit equilateral triangles spanned by a finite point set.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Finset, EuclideanSpace, Real, Fin
Proof uses: Finset.card, Finset.filter, Erdos755.IsUnitEquilateralTriangle, Classical.propDecidable, Finset.powersetCard, OfNat.ofNat
1 instances, coercions and auxiliary constants left out.
```
noncomputable def unitEquilateralTriangleCount (d : ℕ)
    (P : Finset (EuclideanSpace ℝ (Fin d))) : ℕ :=
…
```
def Erdos755.anySizeEquilateralTriangleCount Number of equilateral triangles of any positive side length spanned by a finite point set line 62 source
Source documentation
Number of equilateral triangles of any positive side length spanned by a finite point set.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Finset, EuclideanSpace, Real, Fin
Proof uses: Finset.card, Finset.filter, Erdos755.IsAnySizeEquilateralTriangle, Classical.propDecidable, Finset.powersetCard, OfNat.ofNat
1 instances, coercions and auxiliary constants left out.
```
noncomputable def anySizeEquilateralTriangleCount (d : ℕ)
    (P : Finset (EuclideanSpace ℝ (Fin d))) : ℕ :=
…
```
def Erdos755.TUnit Maximum number of unit equilateral triangles spanned by $n$ points in $\mathbb{R}^d$ line 68 source
Source documentation
Maximum number of unit equilateral triangles spanned by $n$ points in $\mathbb{R}^d$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: SupSet.sSup, Set.ofPred, Exists, Finset, EuclideanSpace, Real, Fin, And, Eq, Finset.card, Erdos755.unitEquilateralTriangleCount
1 instances, coercions and auxiliary constants left out.
```
noncomputable def TUnit (d n : ℕ) : ℕ :=
…
```
def Erdos755.TAnySize Maximum number of equilateral triangles of any size spanned by $n$ points in $\mathbb{R}^d$ line 73 source
Source documentation
Maximum number of equilateral triangles of any size spanned by $n$ points in $\mathbb{R}^d$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: SupSet.sSup, Set.ofPred, Exists, Finset, EuclideanSpace, Real, Fin, And, Eq, Finset.card, Erdos755.anySizeEquilateralTriangleCount
1 instances, coercions and auxiliary constants left out.
```
noncomputable def TAnySize (d n : ℕ) : ℕ :=
…
```
theorem Erdos755.erdos_755.test_dim_one line 78 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, Erdos755.unitEquilateralTriangleCount, OfNat.ofNat, EmptyCollection.emptyCollection, Finset, EuclideanSpace, Real, Fin
Proof uses: of_eq_true, Finset.card, Finset.filter, Erdos755.IsUnitEquilateralTriangle, Classical.propDecidable, Finset.powersetCard, Eq.trans, True, Membership.mem, Not, forall_congr, And, implies_congr, LE.le, congrFun', congrArg, Eq.refl, False
29 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos755.erdos_755 sorry line 93 source
depends on sorry what this means
Statement uses: Iff, True, Exists, Nat, Real, And, Asymptotics.IsLittleO, Filter.atTop, OfNat.ofNat, Filter.Eventually, LE.le, Erdos755.TUnit, HMul.hMul, HAdd.hAdd, HDiv.hDiv, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
44 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos755.erdos_755.variants.any_size_cdl Clemen, Dumitrescu, and Liu [CDL25b] proved the stronger version where equilateral triangles of all positive side lengths are counted sorry line 105 source
Source documentation
Clemen, Dumitrescu, and Liu [CDL25b] proved the stronger version where equilateral triangles of all positive side lengths are counted.

depends on sorry what this means
Statement uses: Asymptotics.IsEquivalent, Nat, Real, Filter.atTop, Erdos755.TAnySize, OfNat.ofNat, HMul.hMul, HDiv.hDiv, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
46 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_755.variants.any_size_cdl :
    (fun n : ℕ => (TAnySize 6 n : ℝ)) ~[atTop]
      (fun n : ℕ => (1 / 27 : ℝ) * (n : ℝ) ^ 3) := by
…
```
Mentions: Erdos755.TAnySize