103.lean`

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

6 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 103

References:

erdosproblems.com/103

Imports (1)

FormalConjecturesUtil

def Erdos103.separatedSets The sets of $n$ points in $\mathbb{R}^2$ whose pairwise distances are all at least $1$ line 34 source
Source documentation
The sets of $n$ points in $\mathbb{R}^2$ whose pairwise distances are all at least $1$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Set, Finset, EuclideanSpace, Real, Fin, OfNat.ofNat
Proof uses: Set.ofPred, And, Eq, Finset.card, Metric.IsSeparated', ENNReal
17 instances, coercions and auxiliary constants left out.
```
def separatedSets (n : ℕ) : Set (Finset ℝ²) :=
…
```
def Erdos103.IsMinimiser A is a minimiser for n if it is a separated set of n points whose diameter is at most the diameter of every separated set of n points (compare Erdos99.erdos_99) line 39 source
Source documentation
A is a minimiser for n if it is a separated set of n points whose diameter is at most the diameter of every separated set of n points (compare Erdos99.erdos_99).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Finset, EuclideanSpace, Real, Fin, OfNat.ofNat
Proof uses: And, Membership.mem, Set, Erdos103.separatedSets, IsMinOn, Metric.diam, ENNReal
15 instances, coercions and auxiliary constants left out.
```
def IsMinimiser (n : ℕ) (A : Finset ℝ²) : Prop :=
…
```
def Erdos103.Congruent Two finite sets of points are congruent if an isometry of the plane maps one onto the other line 45 source
Source documentation
Two finite sets of points are congruent if an isometry of the plane maps one onto the other.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Finset, EuclideanSpace, Real, Fin, OfNat.ofNat, Nat
Proof uses: Exists, IsometryEquiv, ENNReal, Eq, Set, Set.image
18 instances, coercions and auxiliary constants left out.
```
def Congruent (A B : Finset ℝ²) : Prop :=
…
```
def Erdos103.h Documented declaration: h line 51 source
Source documentation
h n is the number of congruence classes of minimisers for n points, as an element of ℕ∞: the supremum of the cardinalities of finite families of pairwise incongruent minimisers, which is ⊤ when there are infinitely many classes.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, ENat
Proof uses: iSup, Finset, EuclideanSpace, Real, Fin, OfNat.ofNat, And, Membership.mem, Erdos103.IsMinimiser, Ne, Not, Erdos103.Congruent, Finset.card
10 instances, coercions and auxiliary constants left out.
```
noncomputable def h (n : ℕ) : ℕ∞ :=
…
```
theorem Erdos103.erdos_103 Let $h(n)$ count the number of incongruent sets of $n$ points in $\mathbb{R}^2$ which minimise the diameter subject to the constraint that $d(x,y)\geq 1$ for all points $x\neq y$ sorry line 61 source
Source documentation
Let $h(n)$ count the number of incongruent sets of $n$ points in $\mathbb{R}^2$ which minimise the diameter subject to the constraint that $d(x,y)\geq 1$ for all points $x\neq y$. Is it true that $h(n)\to \infty$?

depends on sorry what this means
Statement uses: Iff, True, Filter.Tendsto, Nat, ENat, Erdos103.h, Filter.atTop, nhds, Top.top
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
7 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_103 : answer(sorry) ↔ Tendsto h atTop (nhds ⊤) := by
…
```
Mentions: Erdos103.h
theorem Erdos103.erdos_103.variants.two_le It is not even known whether $h(n)\geq 2$ for all large $n$ sorry line 68 source
Source documentation
It is not even known whether $h(n)\geq 2$ for all large $n$.

depends on sorry what this means
Statement uses: Iff, True, Filter.Eventually, Nat, LE.le, ENat, OfNat.ofNat, Erdos103.h, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
14 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_103.variants.two_le : answer(sorry) ↔ ∀ᶠ n in atTop, 2 ≤ h n := by
…
```
Mentions: Erdos103.h