Vela Mathematics Program: problems

Problem ledger

55 problems · 1/2Number SourcesGraph view

declared open · formalized · $500erdos:89

(fun n => ↑n / √(Real.log ↑n)) =O[Filter.atTop] fun n => ↑(EuclideanGeometry.minimalDistinctDistances n)

geometrydistancesA186704 · A1316282 sources

declared disproved (Lean) · formalized · $500erdos:90

Conjectured upper bound on how many pairs among

n

points in the plane can be exactly one unit apart.

geometrydistancesA1867055 sources

declared open · formalizederdos:91

(∀ᶠ (n : ℕ) in Filter.atTop, ¬Erdos91.UniqueMinimizer n) ↔ True

geometrydistancesA186704 · possible3 sources

declared disproved · formalized · $500erdos:92

sorry ↔ ∃ c > 0, ∀ᶠ (n : ℕ) in Filter.atTop, ↑(Erdos92.f n) ≤ ↑n ^ (c / Real.log (Real.log ↑n))

geometrydistancespossible2 sources

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 · $500erdos:95

Erdős problem 95

geometryconvexdistancespossible

declared open · formalizederdos:96

∀ (n : ℕ), BddAbove (Erdos96.convexUnitDistanceCounts n)

geometrydistancesconvexpossible2 sources

declared falsifiable · formalized · $100erdos:97

True ↔ ∀ (A : Finset (EuclideanSpace ℝ (Fin 2))), A.Nonempty → EuclideanGeometry.ConvexIndep ↑A → ¬Erdos97.HasNEquidistantProperty 4 A

geometrydistancesconvexN/A2 sources

declared open · formalizederdos:98

True ↔ Filter.Tendsto (fun n => ↑(Erdos98.h n) / ↑n) Filter.atTop Filter.atTop

geometrydistancespossible2 sources

declared open · formalized · $100erdos:99

True ↔ ∀ᶠ (n : ℕ) in Filter.atTop, ∀ (A : Finset (EuclideanSpace ℝ (Fin 2))), A.card = n → Erdos99.HasMinDist1 A → IsMinOn (fun B => Metric.diam ↑B) {B | B.card = n ∧ Erdos99.HasMinDist1 B} A → ∃ p ∈ A, ∃ q ∈ A, ∃ r ∈ A, Erdos99.FormsEquilateralTriangle p q r

geometrydistancesN/A2 sources

declared open · formalizederdos:100

True ↔ ∃ C > 0, ∀ᶠ (n : ℕ) in Filter.atTop, ∀ (A : Finset (EuclideanSpace ℝ (Fin 2))), A.card = n → Erdos100.DistancesSeparated A → Metric.diam ↑A > C * ↑n

geometrydistancesN/A2 sources

declared openerdos:103

Erdős problem 103

geometrydistancespossible

declared open · $100erdos:132

Erdős problem 132

distancesN/A

declared disproved · $250erdos:135

Erdős problem 135

distancesgeometrypossible

declared open · formalizederdos:212

True ↔ ∃ u, Dense u ∧ u.Pairwise fun c₁ c₂ => dist c₁ c₂ ∈ Set.range Rat.cast

geometrydistancesN/A2 sources

declared open · formalizederdos:213

True ↔ ∀ n ≥ 4, Erdos213.Erdos213For n

geometrydistancesN/A2 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 openerdos:217

Erdős problem 217

geometrydistancespossible

declared solvederdos:223

Erdős problem 223

geometrydistancespossible

declared provederdos:232

Erdős problem 232

geometrydistancesN/A

declared solved (Lean) · formalizederdos:502

∀ (n : ℕ) (A : Set (EuclideanSpace ℝ (Fin n))), A.Finite → {d | ∃ x ∈ A, ∃ y ∈ A, x ≠ y ∧ dist x y = d}.ncard = 2 → A.ncard ≤ (n + 2).choose 2

geometrydistancesA0276275 sources

declared open · formalizederdos:503

∀ (n : ℕ), IsGreatest {x | ∃ A, ∃ (_ : A.IsIsosceles), A.ncard = x} sorry

geometrydistancesA1757693 sources

declared open · $500erdos:604

Erdős problem 604

geometrydistancespossible

declared provederdos:605

Erdős problem 605

geometrydistancespossible

declared provederdos:652

Erdős problem 652

geometrydistancesN/A2 sources

declared open · formalizederdos:653

True ↔ ∃ o, o =o[Filter.atTop] 1 ∧ ∀ᶠ (n : ℕ) in Filter.atTop, (1 - o n) * ↑n ≤ ↑(EuclideanGeometry.maximalDistinctDistancesFrom n)

geometrydistancespossible2 sources

declared openerdos:654

n

planar points have no four concyclic, must some point determine

(1 - o(1))n

distinct distances? Failing that, can one always force more than

(1/3 + c)n

geometrydistancespossible3 sources

declared open · formalizederdos:655

False ↔ ∃ c > 0, ∀ᶠ (n : ℕ) in Filter.atTop, ∀ (X : Finset (EuclideanSpace ℝ (Fin 2))), X.card = n → Erdos655.IsValid X → (1 + c) * ↑n / 2 ≤ ↑(EuclideanGeometry.distinctDistances X)

geometrydistancespossible3 sources

declared openerdos:657

Erdős problem 657

geometrydistancespossible

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 openerdos:660

Erdős problem 660

geometrydistancesconvexpossible

declared open · $50erdos:661

Erdős problem 661

geometrydistancespossible

declared openerdos:662

Among sufficiently large one-separated planar point sets, does the triangular lattice maximize the number of distances below each threshold? Explicit rational oblique lattices beat the triangular lattice under several closed- and strict-shell readings.

geometrydistancesN/A3 sources

declared openerdos:668

Erdős problem 668

geometrydistancesA385657

declared openerdos:670

Erdős asked whether every

n

-point set in Euclidean space whose pairwise distances are mutually at least 1 apart must have diameter at least

(1+o(1))n^2

. Disproved: an explicit high-dimensional construction beats the conjectured constant.

geometrydistancesN/A2 sources

declared provederdos:754

Erdős problem 754

geometrydistancespossible

declared proved (Lean) · formalizederdos:756

True ↔ (fun n => ↑n) =O[Filter.atTop] fun n => ↑(Erdos756.maxRichDistances n)

geometrydistancespossible5 sources

declared open · formalizederdos:757

∀ {A : Set ℝ}, sorry = sSup {c | Erdos757.IsAdmissible c}

geometrydistancessidon setspossible2 sources

declared openerdos:953

What is the largest possible measure of a subset of a radius-

R

disk in

\mathbb{R}^2

containing no pair of points at a positive integer distance? A Poisson-Bessel kernel argument gives

M(R) \ll R^{1/2}

; with Sárközy's lower construction,

M(R) = R^{1/2 + o(1)}

geometrydistancesN/A3 sources

declared openerdos:956

Erdős problem 956

geometrydistancesconvexpossible2 sources

declared provederdos:957

Erdős problem 957

geometrydistancesN/A

declared disproved (Lean) · formalizederdos:958

False ↔ ∀ (n : ℕ) (A : Finset (EuclideanSpace ℝ (Fin 2))), A.card = n → (EuclideanGeometry.distanceSet A).card = n - 1 ∧ Finset.image (EuclideanGeometry.distanceMultiplicity A) (EuclideanGeometry.distanceSet A) = Finset.Icc 1 (n - 1) → Erdos958.IsEquidistantOnLine A ∨ Erdos958.IsEquidistantOnCircle A

distancesgeometryN/A5 sources

declared open · formalizederdos:959

How large can the difference between the largest and second-largest distance multiplicities be among

n

planar points?

geometrydistancesN/A4 sources

declared falsifiable · formalizederdos:982

∀ (n : ℕ), 3 ≤ n → ∀ (p : Fin n → EuclideanSpace ℝ (Fin 2)), Function.Injective p → EuclideanGeometry.IsConvexPolygon p → ∃ i, {d | ∃ j, j ≠ i ∧ d = dist (p i) (p j)}.ncard ≥ n / 2

geometryconvexdistancesA0045262 sources

declared falsifiable · formalizederdos:1082

True ↔ ∀ (A : Finset (EuclideanSpace ℝ (Fin 2))), EuclideanGeometry.NonTrilinear ↑A → A.card / 2 ≤ EuclideanGeometry.distinctDistances A

geometrydistancespossible3 sources

declared openerdos:1083

Erdős problem 1083

geometrydistancesA186704 · possible

declared open · formalizederdos:1084

∀ {n : ℕ}, n ≠ 0 → Erdos1084.f 2 n < 3 * n

geometrydistancesA045945 · possible3 sources

declared open · formalizederdos:1085

∃ c > 0, ∀ᶠ (n : ℕ) in Filter.atTop, ↑n ^ (1 + c / Real.log (Real.log ↑n)) < ↑(Erdos1085.f 2 n)

geometrydistancesA186705 · possible2 sources

declared openerdos:1086

Erdős problem 1086

geometrydistancespossible

Problems · Vela Mathematics Program

Problem ledger