96.lean`

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

4 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 96

Reference: erdosproblems.com/96

Imports (1)

FormalConjecturesUtil

def Erdos96.convexUnitDistanceCounts The set of all possible numbers of unit distances determined by the vertices of a convex $n$-gon line 39 source
Source documentation
The set of all possible numbers of unit distances determined by the vertices of a convex $n$-gon.

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, EuclideanGeometry.ConvexIndep, unitDistNum, ENNReal
12 instances, coercions and auxiliary constants left out.
```
noncomputable def convexUnitDistanceCounts (n : ℕ) : Set ℕ :=
…
```
theorem Erdos96.convexUnitDistanceCounts_bddAbove This lemma confirms that the set of possible unit-distance counts is bounded above, which ensures that taking the supremum (sSup) is a well-defined operation line 48 source
Source documentation
This lemma confirms that the set of possible unit-distance counts is bounded above, which ensures that taking the supremum (sSup) is a well-defined operation. The trivial upper bound is the total number of pairs of points, $\binom{n}{2}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, BddAbove, Erdos96.convexUnitDistanceCounts
Proof uses: Exists.intro, Membership.mem, Set, upperBounds, Nat.choose, OfNat.ofNat, Finset, EuclideanSpace, Real, Fin, Exists, Eq, Finset.card, EuclideanGeometry.ConvexIndep, unitDistNum, ENNReal, LE.le, unitDistNum_le_choose_two
17 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
theorem convexUnitDistanceCounts_bddAbove (n : ℕ) : BddAbove <| convexUnitDistanceCounts n := by
…
```
Mentions: Erdos96.convexUnitDistanceCounts
def Erdos96.maxConvexUnitDistances The maximum number of unit distances determined by the vertices of a convex $n$-gon line 57 source
Source documentation
The maximum number of unit distances determined by the vertices of a convex $n$-gon. This function is often denoted as $U_c(n)$ in combinatorics.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: SupSet.sSup, Erdos96.convexUnitDistanceCounts
1 instances, coercions and auxiliary constants left out.
```
noncomputable def maxConvexUnitDistances (n : ℕ) : ℕ :=
…
```
theorem Erdos96.erdos_96 If $n$ points in $\mathbb{R}^2$ form a convex polygon then there are $O(n)$ many pairs which are distance $1$ apart sorry line 65 source
Source documentation
If $n$ points in $\mathbb{R}^2$ form a convex polygon then there are $O(n)$ many pairs which are distance $1$ apart.

depends on sorry what this means
Statement uses: Iff, True, Asymptotics.IsBigO, Nat, Real, Filter.atTop, Erdos96.maxConvexUnitDistances
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
9 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_96 :
    answer(sorry) ↔ (fun n => (maxConvexUnitDistances n : ℝ)) =O[atTop] fun n => (n : ℝ) := by
…
```
Mentions: Erdos96.maxConvexUnitDistances