1088.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 1088

Reference: erdosproblems.com/1088

Imports (1)

FormalConjecturesUtil

def Erdos1088.PairwiseDistancesDistinct A finite set of points has all pairwise distances distinct line 36 source
Source documentation
A finite set of points has all pairwise distances distinct.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Finset, EuclideanSpace, Real, Fin
Proof uses: Eq, distinctDistances, OfNat.ofNat, ENNReal, Nat.choose, Finset.card
10 instances, coercions and auxiliary constants left out.
```
def PairwiseDistancesDistinct (A : Finset (ℝ^d)) : Prop :=
…
```
def Erdos1088.HasSubsetWithDistinctDistances The finite set S contains an n-point subset with all pairwise distances distinct line 40 source
Source documentation
The finite set S contains an n-point subset with all pairwise distances distinct.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Finset, EuclideanSpace, Real, Fin
Proof uses: Exists, And, LE.le, Eq, Finset.card, Erdos1088.PairwiseDistancesDistinct
3 instances, coercions and auxiliary constants left out.
```
def HasSubsetWithDistinctDistances (n : ℕ) (S : Finset (ℝ^d)) : Prop :=
…
```
def Erdos1088.cardSet The set of cardinalities m such that every m-point subset of ℝ^d contains an n-point subset with all pairwise distances distinct line 47 source
Source documentation
The set of cardinalities m such that every m-point subset of ℝ^d contains an n-point subset with all pairwise distances distinct.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Set
Proof uses: Set.ofPred, And, LE.le, Finset, EuclideanSpace, Real, Fin, Eq, Finset.card, Erdos1088.HasSubsetWithDistinctDistances
1 instances, coercions and auxiliary constants left out.
```
def cardSet (d n : ℕ) : Set ℕ :=
…
```
def Erdos1088.f The least m such that every m-point subset of ℝ^d contains an n-point subset with all pairwise distances distinct line 54 source
Source documentation
The least m such that every m-point subset of ℝ^d contains an n-point subset with all pairwise distances distinct.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: InfSet.sInf, Erdos1088.cardSet
1 instances, coercions and auxiliary constants left out.
```
noncomputable def f (d n : ℕ) : ℕ := sInf (cardSet d n)
…
```
Mentions: Erdos1088.cardSet
theorem Erdos1088.f_isLeast f d n is the least cardinality with the required distinct-distance property sorry line 58 source
Source documentation
f d n is the least cardinality with the required distinct-distance property.

depends on sorry what this means
Statement uses: Nat, IsLeast, Erdos1088.cardSet, Erdos1088.f
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem f_isLeast (d n : ℕ) : IsLeast (cardSet d n) (f d n) := by
…
```
Mentions: Erdos1088.cardSet, Erdos1088.f
theorem Erdos1088.erdos_1088 Let $f_d(n)$ be the minimal $m$ such that any set of $m$ points in $\mathbb{R}^d$ contains a set of $n$ points for which any two determined distances are distinct sorry line 73 source
Source documentation
Let $f_d(n)$ be the minimal $m$ such that any set of $m$ points in $\mathbb{R}^d$ contains a set of $n$ points for which any two determined distances are distinct. Erdős Problem 1088 asks to estimate $f_d(n)$. In particular, is it true that, for every fixed $n \geq 3$, $$ f_d(n) = 2^{o(d)} $$ as $d \to \infty$?
The little-$o$ condition is stated after taking the base-$2$ logarithm.

depends on sorry what this means
Statement uses: Iff, True, Nat, GE.ge, OfNat.ofNat, Asymptotics.IsLittleO, Real, Filter.atTop, Real.logb, Erdos1088.f
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_1088 :
    answer(sorry) ↔
      ∀ n ≥ 3,
        (fun d : ℕ ↦ Real.logb 2 (f d n : ℝ)) =o[atTop] (fun d : ℕ ↦ (d : ℝ)) := by
…
```
Mentions: Erdos1088.f