92.lean`

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7 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 92

Both questions here are disproved, by way of Erdős Problem 90. The source says so directly: this is a stronger form of the unit distance conjecture, so the disproof of that conjecture disproves these too. See FormalConjectures.ErdosProblems.«90».

Reference: erdosproblems.com/92

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/Subsets/FC100OpenSet1.lean

def Erdos92.maxEquidistantPointsAt For a given point x and a set of other points, this function finds the maximum number of points that lie on a single circle centered at x line 42 source
Source documentation
For a given point x and a set of other points, this function finds the maximum number of points that lie on a single circle centered at x. It does this by grouping the other points by their distance to x and finding the size of the largest group.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: EuclideanSpace, Real, Fin, OfNat.ofNat, Nat, Finset
Proof uses: SupSet.sSup, Finset.image, Finset.card, Finset.filter, Eq, Dist.dist, ENNReal, Finset.erase
18 instances, coercions and auxiliary constants left out.
```
noncomputable def maxEquidistantPointsAt (x : ℝ²) (points : Finset ℝ²) : ℕ :=
…
```
def Erdos92.hasMinEquidistantProperty This property holds for a set of points A if every point x in A has at least k other points from A that are equidistant from x line 51 source
Source documentation
This property holds for a set of points A if every point x in A has at least k other points from A that are equidistant from x.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Finset, EuclideanSpace, Real, Fin, OfNat.ofNat
Proof uses: And, Finset.Nonempty, Membership.mem, LE.le, Erdos92.maxEquidistantPointsAt
5 instances, coercions and auxiliary constants left out.
```
def hasMinEquidistantProperty (k : ℕ) (A : Finset ℝ²) : Prop :=
…
```
def Erdos92.possible_f_values The set of all possible values k for which there exists a set of n points satisfying the hasMinEquidistantProperty k line 58 source
Source documentation
The set of all possible values k for which there exists a set of n points satisfying the hasMinEquidistantProperty k. The function f(n) will be the supremum of this set.

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, Eq, Finset.card, Erdos92.hasMinEquidistantProperty
1 instances, coercions and auxiliary constants left out.
```
noncomputable def possible_f_values (n : ℕ) : Set ℕ :=
…
```
theorem Erdos92.possible_f_values_BddAbove A sanity check to ensure the set of possible f(n) values is bounded above line 67 source
Source documentation
A sanity check to ensure the set of possible f(n) values is bounded above. A trivial bound is n, since the points equidistant from any x form a subset of the other n - 1 points. This ensures sSup is well-defined.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, BddAbove, Erdos92.possible_f_values
Proof uses: Exists.intro, Membership.mem, Set, upperBounds, Finset, EuclideanSpace, Real, Fin, OfNat.ofNat, Exists, Eq, Finset.card, Erdos92.hasMinEquidistantProperty, LE.le, Finset.Nonempty, Erdos92.maxEquidistantPointsAt, LE.le.trans, id, csSup_le', Finset.image and 13 more
34 instances, coercions and auxiliary constants left out.

Tactic trace: 8 steps
Open to load the checked before-and-after goal states.
```
theorem possible_f_values_BddAbove (n : ℕ) : BddAbove (possible_f_values n) := by
…
```
Mentions: Erdos92.possible_f_values
def Erdos92.f Let $f(n)$ be maximal such that there exists a set $A$ of $n$ points in $\mathbb^2$ in which every $x \in A$ has at least $f(n)$ points in $A$ equidistant from $x$ line 84 source
Source documentation
Let $f(n)$ be maximal such that there exists a set $A$ of $n$ points in $\mathbb^2$ in which every $x \in A$ has at least $f(n)$ points in $A$ equidistant from $x$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: SupSet.sSup, Erdos92.possible_f_values
1 instances, coercions and auxiliary constants left out.
```
noncomputable def f (n : ℕ) : ℕ := sSup <| possible_f_values n
…
```
Mentions: Erdos92.possible_f_values
theorem Erdos92.erdos_92.variants.weak Is it true that $f(n)\leq n^{o(1)}$? sorry line 95 source
Source documentation
Is it true that $f(n)\leq n^{o(1)}$?
The source records this as disproved: "This is a stronger form of the unit distance conjecture (see [90]). As such the recent disproof of [90] also disproves this." That disproof is Erdos90.erdos_90, which this repository already records as research solved with the answer False.

depends on sorry what this means
Statement uses: Iff, False, Exists, Nat, Real, And, Asymptotics.IsLittleO, Filter.atTop, OfNat.ofNat, LE.le, Erdos92.f, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
21 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_92.variants.weak : answer(False) ↔ ∃ o : ℕ → ℝ,
  o =o[atTop] (1 : ℕ → ℝ) ∧ ∀ n, (f n : ℝ) ≤ n^(o n) := by
…
```
Mentions: Erdos92.f
theorem Erdos92.erdos_92.variants.strong Or even $f(n) < n^{c/\log\log n}$ for some constant $c > 0$? sorry line 107 source
Source documentation
Or even $f(n) < n^{c/\log\log n}$ for some constant $c > 0$?
Also disproved, and by the weak form rather than separately: since $c/\log\log n \to 0$, the bound $n^{c/\log\log n}$ is of the form $n^{o(1)}$, so this statement implies erdos_92.variants.weak and is false whenever that one is.

depends on sorry what this means
Statement uses: Iff, False, Exists, Real, And, GT.gt, OfNat.ofNat, Filter.Eventually, Nat, LE.le, Erdos92.f, HPow.hPow, HDiv.hDiv, Real.log, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
25 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_92.variants.strong : answer(False) ↔
    ∃ c > 0, ∀ᶠ n in atTop, (f n : ℝ) ≤ n^(c / (n : ℝ).log.log) := by
…
```
Mentions: c, Erdos92.f