104.lean`

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

4 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 104

References:

erdosproblems.com/104
[El84] Elekes, G., {$n$} points in the plane can determine $n^{3/2}$ unit circles. Combinatorica (1984), 131.
[Er75h] Erdős, P., Some problems on elementary geometry. Austral. Math. Soc. Gaz. (1975), 2-3.
[Er81d] Erdős, P., Some applications of graph theory and combinatorial methods to number theory and geometry. Algebraic methods in graph theory, Vol. I, II (Szeged, 1978) (1981), 137-148.
[Er92e] Erdős, Pál, Some Unsolved problems in Geometry, Number Theory and Combinatorics. Eureka (1992), 44-48.
[HaMe86] Harborth, Heiko and Mengersen, Ingrid, Point sets with many unit circles. Discrete Math. (1986), 193--197.

Imports (1)

FormalConjecturesUtil

def Erdos104.unitCircleCount The number of distinct unit circles containing at least three points of P line 46 source
Source documentation
The number of distinct unit circles containing at least three points of P.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Finset, EuclideanSpace, Real, Fin, OfNat.ofNat, Nat
Proof uses: Set.ncard, EuclideanGeometry.Sphere, ENNReal, Set.ofPred, And, Eq, EuclideanGeometry.Sphere.radius, LE.le, Membership.mem, Set
18 instances, coercions and auxiliary constants left out.
```
noncomputable def unitCircleCount (P : Finset ℝ²) : ℕ :=
…
```
def Erdos104.possibleUnitCircleCounts The set of unit-circle counts attained by configurations of n points in the plane line 50 source
Source documentation
The set of unit-circle counts attained by configurations of n points in the plane.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Set
Proof uses: Set.ofPred, Exists, Finset, EuclideanSpace, Real, Fin, OfNat.ofNat, And, Eq, Finset.card, Erdos104.unitCircleCount
1 instances, coercions and auxiliary constants left out.
```
noncomputable def possibleUnitCircleCounts (n : ℕ) : Set ℕ :=
…
```
def Erdos104.maxUnitCircleCount The maximum number of qualifying unit circles attained by a configuration of n points line 54 source
Source documentation
The maximum number of qualifying unit circles attained by a configuration of n points.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: SupSet.sSup, Erdos104.possibleUnitCircleCounts
1 instances, coercions and auxiliary constants left out.
```
noncomputable def maxUnitCircleCount (n : ℕ) : ℕ :=
…
```
theorem Erdos104.erdos_104 Given $n$ points in $\mathbb{R}^2$ the number of distinct unit circles containing at least three points is $o(n^2)$ sorry line 61 source
Source documentation
Given $n$ points in $\mathbb{R}^2$ the number of distinct unit circles containing at least three points is $o(n^2)$.

depends on sorry what this means
Statement uses: Asymptotics.IsLittleO, Nat, Real, Filter.atTop, Erdos104.maxUnitCircleCount, HPow.hPow, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_104 :
    (fun n : ℕ => (maxUnitCircleCount n : ℝ)) =o[atTop] (fun n : ℕ => (n : ℝ) ^ 2) := by
…
```
Mentions: Erdos104.maxUnitCircleCount