1085.lean`

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8 declarations, 7 contain sorry, 0 declared as axioms

Erdős Problem 1085

Let f_d(n) be minimal such that, in any set of n points in ℝ^d, there exist at most f_d(n) pairs of points which are distance 1 apart. Estimate f_d(n).

Reference: erdosproblems.com/1085

Imports (1)

FormalConjecturesUtil

def Erdos1085.f The maximal number of pairs of points which are distance 1 apart that a set of n points in ℝ^d make line 39 source
Source documentation
The maximal number of pairs of points which are distance 1 apart that a set of n points in ℝ^d make.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: iSup, Finset, EuclideanSpace, Real, Fin, Eq, Finset.card, unitDistNum, OfNat.ofNat, ENNReal
10 instances, coercions and auxiliary constants left out.
```
noncomputable def f (d n : ℕ) : ℕ := ⨆ (s : Finset (ℝ^ d)) (_ : s.card = n), unitDistNum s
…
```
Mentions: unitDistNum
theorem Erdos1085.erdos_1085.variants.lower_d2 Erdős showed $f_2(n) > n^{1+c/\log\log n}$ for some $c > 0$ sorry line 45 source
Source documentation
Erdős showed $f_2(n) > n^{1+c/\log\log n}$ for some $c > 0$.

depends on sorry what this means
Statement uses: Exists, Real, And, GT.gt, OfNat.ofNat, Filter.Eventually, Nat, LT.lt, HPow.hPow, HAdd.hAdd, HDiv.hDiv, Real.log, Erdos1085.f, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
32 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1085.variants.lower_d2 :
    ∃ c > (0 : ℝ), ∀ᶠ n : ℕ in atTop, (n : ℝ) ^ (1 + c / log (log n)) < f 2 n := by
…
```
Mentions: c, Erdos1085.f
theorem Erdos1085.erdos_1085.variants.upper_d2 Spencer, Szemerédi, and Trotter showed $f_2(n) = O(n^{4/3})$ sorry line 51 source
Source documentation
Spencer, Szemerédi, and Trotter showed $f_2(n) = O(n^{4/3})$.

depends on sorry what this means
Statement uses: Asymptotics.IsBigO, Nat, Real, Filter.atTop, Erdos1085.f, OfNat.ofNat, HPow.hPow, HDiv.hDiv
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1085.variants.upper_d2 : (fun n ↦ (f 2 n : ℝ)) =O[atTop] (fun n ↦ (n : ℝ) ^ (4/3 : ℝ)) := by
…
```
Mentions: Erdos1085.f
theorem Erdos1085.erdos_1085.variants.lower_d3 Erdős showed $f_3(n) = Ω(n^{4/3}\log\log n)$ sorry line 56 source
Source documentation
Erdős showed $f_3(n) = Ω(n^{4/3}\log\log n)$.

depends on sorry what this means
Statement uses: Asymptotics.IsBigO, Nat, Real, Filter.atTop, HMul.hMul, HPow.hPow, HDiv.hDiv, OfNat.ofNat, Real.log, Erdos1085.f
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
30 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1085.variants.lower_d3 :
    (fun n : ℕ ↦ (n : ℝ) ^ (4/3 : ℝ) * log (log n)) =O[atTop] (fun n ↦ (f 3 n : ℝ)) := by
…
```
Mentions: Erdos1085.f
theorem Erdos1085.erdos_1085.variants.upper_d3 Is the $n^{4/3}\log\log n$ lower bound in 3D also an upper bound? sorry line 62 source
Source documentation
Is the $n^{4/3}\log\log n$ lower bound in 3D also an upper bound?.

depends on sorry what this means
Statement uses: Iff, True, Asymptotics.IsBigO, Nat, Real, Filter.atTop, Erdos1085.f, OfNat.ofNat, HMul.hMul, HPow.hPow, HDiv.hDiv, Real.log
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
30 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1085.variants.upper_d3 : answer(sorry) ↔
    (fun n ↦ (f 3 n : ℝ)) =O[atTop] (fun n : ℕ ↦ (n : ℝ) ^ (4/3 : ℝ) * log (log n)) := by
…
```
Mentions: Erdos1085.f
theorem Erdos1085.erdos_1085.variants.lower_d4_lenz Lenz showed that, for $d \ge 4$, $f_d(n) \ge \frac{p - 1}{2p} n^2 - O(1)$ where $p = \lfloor\frac d2\rfloor$ sorry line 69 source
Source documentation
Lenz showed that, for $d \ge 4$, $f_d(n) \ge \frac{p - 1}{2p} n^2 - O(1)$ where $p = \lfloor\frac d2\rfloor$.

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Exists, Real, HSub.hSub, HMul.hMul, HDiv.hDiv, HPow.hPow, Erdos1085.f
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_1085.variants.lower_d4_lenz (hd : 4 ≤ d) :
    ∃ C : ℝ, ∀ n : ℕ, ↑(d / 2 - 1) / (2 * ↑(d / 2)) * n ^ 2 - C ≤ f d n := by
…
```
Mentions: Erdos1085.f
theorem Erdos1085.erdos_1085.variants.upper_d4_erdos Erdős showed that, for $d \ge 4$, $f_d(n) \le \left(\frac{p - 1}{2p} + o(1)\right) n^2$ where $p = \lfloor\frac d2\rfloor$ sorry line 76 source
Source documentation
Erdős showed that, for $d \ge 4$, $f_d(n) \le \left(\frac{p - 1}{2p} + o(1)\right) n^2$ where $p = \lfloor\frac d2\rfloor$.

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Exists, Real, And, Filter.Tendsto, Filter.atTop, nhds, Erdos1085.f, HMul.hMul, HAdd.hAdd, HDiv.hDiv, HSub.hSub, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
56 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1085.variants.upper_d4_erdos (hd : 4 ≤ d) :
    ∃ g : ℕ → ℝ, Tendsto g atTop (𝓝 0) ∧
      ∀ n, f d n ≤ (↑(d / 2 - 1) / (2 * ↑(d / 2)) + g n) * n ^ 2 := by
…
```
Mentions: Erdos1085.f
theorem Erdos1085.erdos_1085.variants.upper_lower_d5_odd Documented declaration: upper_lower_d5_odd sorry line 85 source
Source documentation
Erdős and Pach showed that, for $d \ge 5$ odd, there exist constants $c_1(d), c_2(d) > 0$ such that $\frac{p - 1}{2p} n^2 + c_1 n^{4/3} \le f_d(n) \le \frac{p - 1}{2p} n^2 + c_2 n^{4/3}$ where $p = \lfloor\frac d2\rfloor$.

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Odd, Exists, Real, And, GT.gt, Filter.Eventually, HAdd.hAdd, HMul.hMul, HDiv.hDiv, HSub.hSub, HPow.hPow, Erdos1085.f, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
56 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1085.variants.upper_lower_d5_odd (hd : 5 ≤ d) (hd_odd : Odd d) :
    ∃ c₁ > (0 : ℝ), ∃ c₂ : ℝ, ∀ᶠ n in atTop,
      ↑(d / 2 - 1) / (2 * ↑(d / 2)) * n ^ 2 + c₁ * n ^ (4 / 3 : ℝ) ≤ f d n ∧
      f d n ≤ ↑(d / 2 - 1) / (2 * ↑(d / 2)) * n ^ 2 + c₂ * n ^ (4 / 3 : ℝ) := by
…
```
Mentions: Erdos1085.f