209.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 209

References:

erdosproblems.com/209
[Er84] Erdős, P., Research problems. Period. Math. Hungar. (1984), 101-103.
[ErPu95b] Erdős, Paul and Purdy, George, Extremal problems in combinatorial geometry. Handbook of combinatorics, Vol. 1, 2 (1995), 809-874.
[FuPa84] Füredi, Z. and Palásti, I., Arrangements of lines with a large number of triangles. Proc. Amer. Math. Soc. (1984), 561-566.
[Es16] Escudero, Juan García, Gallai triangles in configurations of lines in the projective plane. C. R. Math. Acad. Sci. Paris (2016), 551-554.

Imports (1)

FormalConjecturesUtil

def Erdos209.pointMultiplicity The number of lines from A that pass through the point p line 41 source
Source documentation
The number of lines from A that pass through the point p.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Finset, AffineSubspace, Real, EuclideanSpace, Fin, OfNat.ofNat, Nat, ENNReal
Proof uses: Set.ncard, Set.ofPred, And, Membership.mem, Set
57 instances, coercions and auxiliary constants left out.
```
noncomputable def pointMultiplicity (A : Finset (AffineSubspace ℝ ℝ²)) (p : ℝ²) : ℕ :=
…
```
def Erdos209.HasGallaiTriangle Documented declaration: HasGallaiTriangle line 49 source
Source documentation
A Gallai triangle (or ordinary triangle) in a collection A of lines: three lines from A which intersect in three points, and each of these intersection points only intersects two lines from A.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Finset, AffineSubspace, Real, EuclideanSpace, Fin, OfNat.ofNat, Nat, ENNReal
Proof uses: Exists, And, Membership.mem, Ne, Eq, Erdos209.pointMultiplicity
55 instances, coercions and auxiliary constants left out.
```
def HasGallaiTriangle (A : Finset (AffineSubspace ℝ ℝ²)) : Prop :=
…
```
theorem Erdos209.erdos_209 Let $A$ be a finite collection of $d\geq 4$ non-parallel lines in $\mathbb{R}^2$ such that there are no points where at least four lines from $A$ meet sorry line 65 source
Source documentation
Let $A$ be a finite collection of $d\geq 4$ non-parallel lines in $\mathbb{R}^2$ such that there are no points where at least four lines from $A$ meet. Must there exist a 'Gallai triangle' (or 'ordinary triangle'): three lines from $A$ which intersect in three points, and each of these intersection points only intersects two lines from $A$?
Füredi and Palásti [FuPa84] showed this is false when $d\geq 4$ is not divisible by $9$. Escudero [Es16] showed this is false for all $d\geq 4$.

depends on sorry what this means
Statement uses: Iff, False, Nat, LE.le, OfNat.ofNat, Finset, AffineSubspace, Real, EuclideanSpace, Fin, ENNReal, Eq, Finset.card, Membership.mem, EuclideanGeometry.IsLine, Set.Pairwise, Not, AffineSubspace.Parallel, Erdos209.pointMultiplicity, Erdos209.HasGallaiTriangle
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
60 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_209 : answer(False) ↔
    ∀ d : ℕ, 4 ≤ d → ∀ A : Finset (AffineSubspace ℝ ℝ²), A.card = d →
      (∀ L ∈ A, IsLine L) →
      ((A : Set (AffineSubspace ℝ ℝ²)).Pairwise fun L₁ L₂ => ¬ L₁ ∥ L₂) →
      (∀ p : ℝ², pointMultiplicity A p ≤ 3) →
      HasGallaiTriangle A := by
…
```
Mentions: Erdos209.pointMultiplicity, Erdos209.HasGallaiTriangle