Vela Mathematics Program: problems

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110 problems · 2/3Number SourcesGraph view

declared openerdos:589

Erdős problem 589

geometrypossible

declared open · $500erdos:604

Erdős problem 604

geometrydistancespossible

declared provederdos:605

Erdős problem 605

geometrydistancespossible

declared solvederdos:606

Erdős problem 606

geometryN/A

declared proved · $250erdos:607

Erdős problem 607

geometrypossible

declared solved · formalized · $25erdos:633

∀ {n : ℕ} {T : Affine.Triangle ℝ (EuclideanSpace ℝ (Fin 2))}, Erdos633.IsCuttable n T → n ≠ 0

geometryN/A2 sources

declared open · $25erdos:634

Erdős problem 634

geometrypossible

declared disprovederdos:651

Erdős problem 651

geometryconvexpossible

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:669

Erdős problem 669

geometryA003035 · A006065 · A008997 · possible

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 openerdos:704

Erdős problem 704

graph theorygeometrychromatic numberN/A

declared provederdos:733

Erdős problem 733

combinatoricsgeometrypossible

declared solvederdos:735

Erdős problem 735

geometryN/A

declared provederdos:754

Erdős problem 754

geometrydistancespossible

declared proved · formalizederdos:755

True ↔ ∃ o, (o =o[Filter.atTop] fun x => 1) ∧ ∀ᶠ (n : ℕ) in Filter.atTop, ↑(Erdos755.TUnit 6 n) ≤ (1 / 27 + o n) * ↑n ^ 3

geometrypossible2 sources

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 open · formalizederdos:769

For the least cutoff

c(n)

after which every

k

occurs as the number of homothetic cubes in a decomposition of the unit

n

-cube, is

c(n) \gg n^n

? The Lean proof shows

c(n) = o(n^n)

along odd dimensions.

number theorygeometryA014544 · possible4 sources

declared proved (Lean) · formalizederdos:798

True ↔ (fun n => ↑(Erdos798.t n)) =o[Filter.atTop] fun n => ↑n

geometryA1164465 sources

declared openerdos:827

Erdős problem 827

geometrypossible

declared openerdos:831

Erdős problem 831

geometrypossible

declared openerdos:838

Erdős problem 838

geometryconvexpossible

declared disproved (Lean) · formalizederdos:846

False ↔ ∀ (A : Set (EuclideanSpace ℝ (Fin 2))), ∀ ε > 0, A.Infinite → Erdos846.NonTrilinearFor A ε → Erdos846.WeaklyNonTrilinear A

geometryN/A6 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 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 disprovederdos:960

Erdős problem 960

geometrypossible3 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 solvederdos:1069

Erdős problem 1069

geometrypossible

declared openerdos:1070

Erdős problem 1070

geometrypossible2 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 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

declared openerdos:1087

Erdős problem 1087

geometrydistancespossible

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