846.lean`

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

3 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 846

Reference: erdosproblems.com/846

Imports (1)

FormalConjecturesUtil

def Erdos846.NonTrilinearFor We say a subset A of points in the plane is ε-non-trilinear if any subset B of A, contains a non-trilinear subset C of size at least ε|B| line 36 source
Source documentation
We say a subset A of points in the plane is ε-non-trilinear if any subset B of A, contains a non-trilinear subset C of size at least ε|B|.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, EuclideanSpace, Real, Fin, OfNat.ofNat, Nat
Proof uses: Finset, LE.le, Exists, And, HMul.hMul, Finset.card, EuclideanGeometry.NonTrilinear, ENNReal
34 instances, coercions and auxiliary constants left out.
```
def NonTrilinearFor (A : Set ℝ²) (ε : ℝ) : Prop :=
…
```
def Erdos846.WeaklyNonTrilinear We say a subset A of points in the plane is weakly non-trilinear if it is a finite union of non-trilinear sets line 42 source
Source documentation
We say a subset A of points in the plane is weakly non-trilinear if it is a finite union of non-trilinear sets.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, EuclideanSpace, Real, Fin, OfNat.ofNat, Nat
Proof uses: Exists, Finset, And, Eq, SupSet.sSup, Membership.mem, EuclideanGeometry.NonTrilinear, ENNReal
27 instances, coercions and auxiliary constants left out.
```
def WeaklyNonTrilinear (A : Set ℝ²) : Prop :=
…
```
theorem Erdos846.erdos_846 Documented declaration: erdos_846 sorry line 57 source
Source documentation
Erdős Problem 846 Let A ⊂ ℝ² be an infinite set for which there exists some ϵ>0 such that in any subset of A of size n there are always at least ϵn with no three on a line. Is it true that A is the union of a finite number of sets where no three are on a line?
In other words, prove or disprove the following statement: every infinite ε-non-trilinear subset of the plane is weakly non-trilinar.

depends on sorry what this means
Statement uses: Iff, False, Set, EuclideanSpace, Real, Fin, OfNat.ofNat, Nat, GT.gt, Set.Infinite, Erdos846.NonTrilinearFor, Erdos846.WeaklyNonTrilinear
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
8 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_846 : answer(False) ↔ ∀ᵉ (A : Set ℝ²) (ε > 0), A.Infinite → NonTrilinearFor A ε →
    WeaklyNonTrilinear A := by
…
```
Mentions: Erdos846.NonTrilinearFor, Erdos846.WeaklyNonTrilinear