507.lean`

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

11 declarations, 7 contain sorry, 0 declared as axioms

Erdős Problem 507

References:

erdosproblems.com/507
[CPZ23] Cohen, Alex, Cosmin Pohoata, and Dmitrii Zakharov. "A new upper bound for the Heilbronn triangle problem." arXiv preprint arXiv:2305.18253 (2023).
[CPZ24] Cohen, Alex, Cosmin Pohoata, and Dmitrii Zakharov. "Lower bounds for incidences." Inventiones mathematicae (2025): 1-74.
[KPS82] Komlós, János, János Pintz, and Endre Szemerédi. "A lower bound for Heilbronn's problem." Journal of the London Mathematical Society 2.1 (1982): 13-24.
[KPS81] Komlós, János, János Pintz, and Endre Szemerédi. "On Heilbronn's triangle problem." Journal of the London Mathematical Society 2.3 (1981): 385-396.

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/GreensOpenProblems/77.lean

def Erdos507.minTriangleArea The minimum (unsigned) area of a triangle determined by three distinct points in a set S line 47 source
Source documentation
The minimum (unsigned) area of a triangle determined by three distinct points in a set S. Collinear triples count with area 0. EuclideanGeometry.triangle_area is a signed area, so the absolute value is taken before the infimum.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Finset, EuclideanSpace, Real, Fin, OfNat.ofNat, Nat
Proof uses: InfSet.sInf, Set.ofPred, Exists, Function.Injective, Membership.mem, Eq, abs, EuclideanGeometry.triangle_area, ENNReal
32 instances, coercions and auxiliary constants left out.
```
noncomputable def minTriangleArea (S : Finset ℝ²) : ℝ :=
…
```
def Erdos507.α $\alpha(n)$ is the supremum of minTriangleArea S over all sets S of $n$ points in the unit disk line 54 source
Source documentation
$\alpha(n)$ is the supremum of minTriangleArea S over all sets S of $n$ points in the unit disk.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real
Proof uses: SupSet.sSup, Set.image, Finset, EuclideanSpace, Fin, OfNat.ofNat, Erdos507.minTriangleArea, Set.ofPred, And, Eq, Finset.card, LE.le, Set, Metric.closedBall, ENNReal, Not, Collinear
43 instances, coercions and auxiliary constants left out.
```
noncomputable def α (n : ℕ) : ℝ :=
…
```
def Erdos507.lowerBest Current best lower bound [KPS82] line 61 source
Source documentation
Current best lower bound [KPS82].

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real
Proof uses: HDiv.hDiv, Real.log, HPow.hPow, OfNat.ofNat
10 instances, coercions and auxiliary constants left out.
```
noncomputable def lowerBest (n : ℕ) : ℝ := Real.log n / (n : ℝ) ^ 2
```
def Erdos507.upperBarrier The "Barrier" function: n^(-7/6) used for the best upper bound [CPZ24] line 66 source
Source documentation
The "Barrier" function: n^(-7/6) used for the best upper bound [CPZ24].

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real
Proof uses: HDiv.hDiv, OfNat.ofNat, HPow.hPow
12 instances, coercions and auxiliary constants left out.
```
noncomputable def upperBarrier (n : ℕ) : ℝ := 1 / (n : ℝ) ^ ((7 : ℝ) / 6)
…
```
theorem Erdos507.erdos_507.equivalent Let $\alpha(n)$ be such that every set of $n$ points in the unit disk contains three points which determine a triangle of area at most $\alpha(n)$ sorry line 73 source
Source documentation
Let $\alpha(n)$ be such that every set of $n$ points in the unit disk contains three points which determine a triangle of area at most $\alpha(n)$. Estimate $\alpha(n)$.

depends on sorry what this means
Statement uses: Asymptotics.IsEquivalent, Nat, Real, Filter.atTop, Erdos507.α, sorryAx, Bool.false
Proof uses: Lean.Name, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
13 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_507.equivalent:
    α ~[atTop] (answer(sorry) : ℕ → ℝ) := by
…
```
Mentions: Erdos507.α
theorem Erdos507.erdos_507.lower Estimate a lower bound for$\alpha(n)$ sorry line 81 source
Source documentation
Estimate a lower bound for$\alpha(n)$.

depends on sorry what this means
Statement uses: Nat, Real, sorryAx, Bool.false, And, Asymptotics.IsLittleO, Filter.atTop, Erdos507.lowerBest, Asymptotics.IsBigO, Erdos507.α
Proof uses: Lean.Name, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_507.lower:
    let ans := (answer(sorry) : ℕ → ℝ)
…
```
theorem Erdos507.erdos_507.upper Estimate an upper bound for$\alpha(n)$ sorry line 90 source
Source documentation
Estimate an upper bound for$\alpha(n)$.

depends on sorry what this means
Statement uses: Nat, Real, sorryAx, Bool.false, And, Asymptotics.IsBigO, Filter.atTop, Erdos507.α, Asymptotics.IsLittleO, Erdos507.upperBarrier
Proof uses: Lean.Name, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_507.upper:
    let ans := (answer(sorry) : ℕ → ℝ)
…
```
theorem Erdos507.erdos_507.variants.upper_trivial It is trivial that $\alpha(n) \ll 1/n$ sorry line 99 source
Source documentation
It is trivial that $\alpha(n) \ll 1/n$.

depends on sorry what this means
Statement uses: Asymptotics.IsBigO, Nat, Real, Filter.atTop, Erdos507.α, HDiv.hDiv, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
19 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_507.variants.upper_trivial : α ≪ (fun n ↦ 1 / (n : ℝ)) := by
…
```
Mentions: Erdos507.α
theorem Erdos507.erdos_507.variants.lower_erdos Erdős observed that $\alpha(n) \gg 1/n^2$ sorry line 106 source
Source documentation
Erdős observed that $\alpha(n) \gg 1/n^2$.

depends on sorry what this means
Statement uses: Asymptotics.IsBigO, Nat, Real, Filter.atTop, HDiv.hDiv, OfNat.ofNat, HPow.hPow, Erdos507.α
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_507.variants.lower_erdos : α ≫ (fun n ↦ 1 / (n : ℝ) ^ 2) := by
…
```
Mentions: Erdos507.α
theorem Erdos507.erdos_507.variants.lower_kps82 Current best lower bound [KPS82] sorry line 113 source
Source documentation
Current best lower bound [KPS82].

depends on sorry what this means
Statement uses: Asymptotics.IsBigO, Nat, Real, Filter.atTop, Erdos507.lowerBest, Erdos507.α
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_507.variants.lower_kps82 : lowerBest ≪ α := by
…
```
Mentions: Erdos507.lowerBest, Erdos507.α
theorem Erdos507.erdos_507.variants.upper_cpz24 Current best upper bound [CPZ24]: $\alpha(n) \ll n^{-7/6 + o(1)}$ sorry line 120 source
Source documentation
Current best upper bound [CPZ24]: $\alpha(n) \ll n^{-7/6 + o(1)}$.

depends on sorry what this means
Statement uses: Exists, Nat, Real, And, Filter.Tendsto, Filter.atTop, nhds, OfNat.ofNat, Asymptotics.IsBigO, Erdos507.α, HMul.hMul, Erdos507.upperBarrier, HPow.hPow
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_507.variants.upper_cpz24 :
    ∃ (o : ℕ → ℝ), Tendsto o atTop (𝓝 0) ∧
    α ≪ (fun n ↦ upperBarrier n * (n : ℝ) ^ o n) := by
…
```
Mentions: Erdos507.α, Erdos507.upperBarrier