94.lean`

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

5 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 94

References:

erdosproblems.com/94
[Er92e] Erdős, Pál, Some Unsolved problems in Geometry, Number Theory and Combinatorics. Eureka (1992), 44-48.
[Er97c] Erdős, Paul, Some of my favorite problems and results. The mathematics of Paul Erdős, I (1997), 47-67.
[LeTh95] Lefmann, Hanno and Thiele, Torsten, Point sets with distinct distances. Combinatorica (1995), 379-408.

Imports (1)

FormalConjecturesUtil

def Erdos94.regularNGon The regular $n$-gon inscribed in the unit circle line 40 source
Source documentation
The regular $n$-gon inscribed in the unit circle.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Finset, EuclideanSpace, Real, Fin, OfNat.ofNat
Proof uses: Finset.image, ENNReal, WithLp.toLp, Matrix.vecCons, Real.cos, HDiv.hDiv, HMul.hMul, Real.pi, Real.sin, Matrix.vecEmpty, Finset.range
19 instances, coercions and auxiliary constants left out.
```
noncomputable def regularNGon (n : ℕ) : Finset ℝ² :=
…
```
theorem Erdos94.erdos_94 Suppose $n$ points in $\mathbb{R}^2$ determine a convex polygon and the set of distances between them is $\{u_1,\ldots,u_t\}$ sorry line 53 source
Source documentation
Suppose $n$ points in $\mathbb{R}^2$ determine a convex polygon and the set of distances between them is $\{u_1,\ldots,u_t\}$. Suppose $u_i$ appears as the distance between $f(u_i)$ many pairs of points. Then $$\sum_i f(u_i)^2 \ll n^3.$$
In [Er97c] Erdős claims that Fishburn solved this, but gives no reference.

depends on sorry what this means
Statement uses: Exists, Real, And, GT.gt, OfNat.ofNat, Finset, EuclideanSpace, Fin, Nat, EuclideanGeometry.ConvexIndep, LE.le, Finset.sum, distanceSet, ENNReal, HPow.hPow, distanceMultiplicity, HMul.hMul, Finset.card
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
50 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_94 : ∃ C > (0 : ℝ), ∀ P : Finset ℝ², ConvexIndep (P : Set ℝ²) →
    ∑ u ∈ distanceSet P, (distanceMultiplicity P u : ℝ) ^ 2 ≤ C * (P.card : ℝ) ^ 3 := by
…
```
Mentions: distanceSet, distanceMultiplicity
theorem Erdos94.erdos_94.variants.sum_multiplicity Note it is trivial that $\sum f(u_i)=\binom{n}{2}$ line 61 source
Source documentation
Note it is trivial that $\sum f(u_i)=\binom{n}{2}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Finset, EuclideanSpace, Real, Fin, OfNat.ofNat, Nat, Eq, Finset.sum, distanceSet, ENNReal, distanceMultiplicity, Nat.choose, Finset.card
Proof uses: sum_distanceMultiplicity
21 instances, coercions and auxiliary constants left out.
```
theorem erdos_94.variants.sum_multiplicity (P : Finset ℝ²) :
    ∑ u ∈ distanceSet P, distanceMultiplicity P u = P.card.choose 2 :=
…
```
Mentions: distanceSet, distanceMultiplicity
theorem Erdos94.erdos_94.variants.no_three_on_a_line Lefmann and Theile [LeTh95] prove a stronger version of this question, that $$\sum_i f(u_i)^2 \ll n^3$$ under the weaker assumption that no three points are on a line sorry line 71 source
Source documentation
Lefmann and Theile [LeTh95] prove a stronger version of this question, that $$\sum_i f(u_i)^2 \ll n^3$$ under the weaker assumption that no three points are on a line.

depends on sorry what this means
Statement uses: Exists, Real, And, GT.gt, OfNat.ofNat, Finset, EuclideanSpace, Fin, Nat, EuclideanGeometry.NonTrilinear, ENNReal, LE.le, Finset.sum, distanceSet, HPow.hPow, distanceMultiplicity, HMul.hMul, Finset.card
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
74 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_94.variants.no_three_on_a_line : ∃ C > (0 : ℝ), ∀ P : Finset ℝ²,
    NonTrilinear (P : Set ℝ²) →
    ∑ u ∈ distanceSet P, (distanceMultiplicity P u : ℝ) ^ 2 ≤ C * (P.card : ℝ) ^ 3 := by
…
```
Mentions: distanceSet, distanceMultiplicity
theorem Erdos94.erdos_94.variants.regular_ngon Erdős and Fishburn also make the stronger conjecture that $\sum f(u_i)^2$ is maximal for the regular $n$-gon (for large enough $n$) sorry line 81 source
Source documentation
Erdős and Fishburn also make the stronger conjecture that $\sum f(u_i)^2$ is maximal for the regular $n$-gon (for large enough $n$).

depends on sorry what this means
Statement uses: Filter.Eventually, Nat, Finset, EuclideanSpace, Real, Fin, OfNat.ofNat, Eq, Finset.card, EuclideanGeometry.ConvexIndep, LE.le, Finset.sum, distanceSet, ENNReal, HPow.hPow, distanceMultiplicity, Erdos94.regularNGon, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
42 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_94.variants.regular_ngon : ∀ᶠ n : ℕ in atTop, ∀ P : Finset ℝ²,
    P.card = n → ConvexIndep (P : Set ℝ²) →
    ∑ u ∈ distanceSet P, (distanceMultiplicity P u : ℝ) ^ 2 ≤
      ∑ u ∈ distanceSet (regularNGon n), (distanceMultiplicity (regularNGon n) u : ℝ) ^ 2 := by
…
```
Mentions: distanceSet, distanceMultiplicity, Erdos94.regularNGon