98.lean`

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

3 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 98

References:

[Er75f] Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108.
[Er83c] Erdős, Paul, Combinatorial problems in geometry. Math. Chronicle (1983), 35-54.
[Er87b] Erdős, P., Some combinatorial and metric problems in geometry. Intuitive geometry (Siófok, 1985) (1987), 167-177.
[Er90] Erdős, Paul, Some of my favourite unsolved problems. A tribute to Paul Erdős (1990), 467-478.
[Er92b] Erdős, Paul, Some of my favourite problems in various branches of combinatorics. Matematiche (Catania) (1992), 231-240.
[EFPR93] Erdős, Paul and Füredi, Zoltán and Pach, János and Ruzsa, Imre Z., The grid revisited. Discrete Math. (1993), 189-196.
[Er94b] Erdős, Paul, Some problems in number theory, combinatorics and combinatorial geometry. Math. Pannon. (1994), 261-269.
[Er97e] Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537.
erdosproblems.com/98

Imports (1)

FormalConjecturesUtil

def Erdos98.h $h(n)$ is the minimum number of distinct distances determined by any $n$-point set in $\mathbb{R}^2$ in general position (no three collinear, no four cocyclic) line 60 source
Source documentation
$h(n)$ is the minimum number of distinct distances determined by any $n$-point set in $\mathbb{R}^2$ in general position (no three collinear, no four cocyclic).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: InfSet.sInf, Set.ofPred, Exists, Finset, EuclideanSpace, Real, Fin, OfNat.ofNat, And, Eq, Finset.card, EuclideanGeometry.InGeneralPosition, distinctDistances, ENNReal
13 instances, coercions and auxiliary constants left out.
```
noncomputable def h (n : ℕ) : ℕ :=
…
```
theorem Erdos98.erdos_98 Let $h(n)$ be such that any $n$ points in $\mathbb{R}^2$, with no three on a line and no four on a circle, determine at least $h(n)$ distinct distances sorry line 70 source
Source documentation
Let $h(n)$ be such that any $n$ points in $\mathbb{R}^2$, with no three on a line and no four on a circle, determine at least $h(n)$ distinct distances. Does $h(n)/n\to \infty$?

depends on sorry what this means
Statement uses: Iff, True, Filter.Tendsto, Nat, Real, HDiv.hDiv, Erdos98.h, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
15 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_98 :
    answer(sorry) ↔ Tendsto (fun n : ℕ ↦ ((h n : ℝ) / (n : ℝ))) atTop atTop := by
…
```
Mentions: Erdos98.h
theorem Erdos98.erdos_98.variants.upper_bound Erdős could not even prove $h(n)\geq n$ sorry line 80 source
Source documentation
Erdős could not even prove $h(n)\geq n$. Pach has shown $h(n) < n^{\log_2 3}$. Erdős, Füredi, and Pach [EFPR93] have improved this to $h(n) < n\exp(c\sqrt{\log n})$ for some constant $c>0$.

depends on sorry what this means
Statement uses: Exists, Real, And, GT.gt, OfNat.ofNat, Filter.Eventually, Nat, LT.lt, Erdos98.h, HMul.hMul, Real.exp, Real.sqrt, Real.log, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
17 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_98.variants.upper_bound :
    ∃ c > (0 : ℝ), ∀ᶠ n : ℕ in atTop,
      (h n : ℝ) < (n : ℝ) * Real.exp (c * Real.sqrt (Real.log (n : ℝ))) := by
…
```
Mentions: c, Erdos98.h