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declared proved (Lean) · formalizederdos:93
∀ (A : Finset (EuclideanSpace ℝ (Fin 2))), EuclideanGeometry.ConvexIndep ↑A → A.card / 2 ≤ EuclideanGeometry.distinctDistances A
geometryconvexdistancesN/A5 sources
declared proved (Lean) · formalized · £25erdos:94
∃ C > 0, ∀ (P : Finset (EuclideanSpace ℝ (Fin 2))), EuclideanGeometry.ConvexIndep ↑P → ∑ u ∈ EuclideanGeometry.distanceSet P, ↑(EuclideanGeometry.distanceMultiplicity P u) ^ 2 ≤ C * ↑P.card ^ 3
geometryconvexdistancesA3878585 sources
declared proved (Lean) · formalizederdos:214
True ↔ ∀ (S : Set (EuclideanSpace ℝ (Fin 2))), Erdos214.UnitDistanceAvoiding S → ∃ p, (∀ (i : Fin 4), p i ∈ Sᶜ) ∧ Congruent p Erdos214.unitSquare
geometrydistancesN/A5 sources
declared proved (Lean) · formalizederdos:224
∀ {d : ℕ} (A : Finset (EuclideanSpace ℝ (Fin d))), A.card = 2 ^ d + 1 → ∃ x y z, x ∈ A ∧ y ∈ A ∧ z ∈ A ∧ x ≠ y ∧ x ≠ z ∧ y ≠ z ∧ Erdos224.ObtuseAt x y z
geometryN/A5 sources
declared proved (Lean) · formalizederdos:353
True ↔ ∀ (A : Set (EuclideanSpace ℝ (Fin 2))), MeasurableSet A → MeasureTheory.volume A = ⊤ → ∃ a ∈ A, ∃ b ∈ A, ∃ c ∈ A, ∃ d ∈ A, EuclideanGeometry.IsIsoscelesTrapezoid a b c d ∧ MeasureTheory.volume ((convexHull ℝ) {a, b, c, d}) = 1
geometryN/A4 sources
declared proved (Lean) · formalizederdos:659
True ↔ ∃ A, (∀ (n : ℕ), (A n).card = n ∧ ∀ S ⊆ A n, S.card = 4 → 3 ≤ EuclideanGeometry.distinctDistances S) ∧ (fun n => ↑(EuclideanGeometry.distinctDistances (A n))) =O[Filter.atTop] fun n => ↑n / √(Real.log ↑n)
geometrydistancespossible6 sources
declared proved (Lean) · formalizederdos:756
True ↔ (fun n => ↑n) =O[Filter.atTop] fun n => ↑(Erdos756.maxRichDistances n)
geometrydistancespossible5 sources
declared proved (Lean) · formalizederdos:798
True ↔ (fun n => ↑(Erdos798.t n)) =o[Filter.atTop] fun n => ↑n
geometryA1164465 sources
declared proved (Lean) · formalizederdos:898
∀ (A B C P L M N : EuclideanSpace ℝ (Fin 2)), AffineIndependent ℝ ![A, B, C] → P ∈ interior ((convexHull ℝ) {A, B, C}) → N ∈ affineSpan ℝ {A, B} → (affineSpan ℝ {P, N}).direction ⟂ (affineSpan ℝ {A, B}).direction → M ∈ affineSpan ℝ {B, C} → (affineSpan ℝ {P, M}).direction ⟂ (affineSpan ℝ {B, C}).direction → L ∈ affineSpan ℝ {C, A} → (affineSpan ℝ {P, L}).direction ⟂ (affineSpan ℝ {C, A}).direction → dist P A + dist P B + dist P C ≥ 2 * (dist P M + dist P N + dist P L)
geometryN/A5 sources
declared proved (Lean) · formalizederdos:1071
True ↔ ∃ S, Maximal (fun T => (∀ seg ∈ T, dist seg.1 seg.2 = 1 ∧ seg.1.ofLp 0 ∈ Set.Icc 0 1 ∧ seg.1.ofLp 1 ∈ Set.Icc 0 1 ∧ seg.2.ofLp 0 ∈ Set.Icc 0 1 ∧ seg.2.ofLp 1 ∈ Set.Icc 0 1) ∧ (↑T).Pairwise Erdos1071.SegmentsDisjoint) S
geometryN/A5 sources
declared proved (Lean) · formalizederdos:1090
True ↔ ∀ (k : ℕ), 3 ≤ k → ∃ A, ∀ (C : ↥A → Fin 2), ∃ S, ∃ (hSA : S ⊆ A), Collinear ℝ ↑S ∧ S.card ≥ k ∧ (∀ y ∈ A, y ∈ affineSpan ℝ ↑S → y ∈ S) ∧ ∃ c, ∀ (x : Fin 2 → ℝ) (hx : x ∈ S), C ⟨x, ⋯⟩ = c
geometryramsey theoryN/A5 sources
declared proved (Lean) · formalizederdos:1121
∀ {n : ℕ} (c : Fin n → EuclideanSpace ℝ (Fin 2)) (r : Fin n → ℝ), (∀ (i : Fin n), 0 < r i) → (∀ (v : EuclideanSpace ℝ (Fin 2)) (t : ℝ), v ≠ 0 → (∀ (i : Fin n), ∀ p ∈ Metric.closedBall (c i) (r i), inner ℝ v p ≠ t) → (∀ (i : Fin n), inner ℝ v (c i) < t) ∨ ∀ (i : Fin n), t < inner ℝ v (c i)) → ∃ z, ⋃ i, Metric.closedBall (c i) (r i) ⊆ Metric.closedBall z (∑ i, r i)
geometryN/A5 sources
Vela Mathematics Program: problems · problems.science