Vela Mathematics Program: problems

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604 problems · 12/13Number SourcesGraph view

declared disproved · formalizederdos:1077

False ↔ ∀ ε > 0, ε < 1 → ∀ α > 0, α < 1 → ∀ᶠ (D : ℝ) in Filter.atTop, ∀ᶠ (n : ℕ) in Filter.atTop, ∀ (G : SimpleGraph (Fin n)), ↑G.edgeSet.ncard > ↑n ^ (1 + α) → ∃ H, H.coe.IsBalanced D ∧ ↑H.verts.ncard > ↑n ^ (1 - α) ∧ ↑H.edgeSet.ncard > ε * ↑H.verts.ncard ^ (1 + α)

graph theoryN/A2 sources

declared disproved (Lean) · formalizederdos:1080

False ↔ ∃ c > 0, ∀ (V : Type) [inst : Fintype V] [Nonempty V] (G : SimpleGraph V) (X Y : Set V), Erdos1080.IsBipartition G X Y → X.ncard = ⌊↑(Fintype.card V) ^ (2 / 3)⌋₊ → ↑G.edgeSet.ncard ≥ c * ↑(Fintype.card V) → ∃ v walk, walk.IsCycle ∧ walk.length = 6

graph theorypossible5 sources

declared falsifiable · formalizederdos:1082

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

geometrydistancespossible3 sources

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 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 disproved · formalizederdos:1092

False ↔ (fun n => ↑n) =o[Filter.atTop] fun n => ↑(Erdos1092.f 2 n)

geometrychromatic numberN/A5 sources

declared open · formalizederdos:1093

sorry ↔ {x | have k := x.1; have n := x.2; 2 * k ≤ n ∧ Erdos1093.deficiency n k = 1 ∧ ∀ (p : ℕ), Nat.Prime p → p ∣ n.choose k → k < p}.Infinite

number theorybinomial coefficientsN/A2 sources

declared open · formalizederdos:1094

{(n, k) | 0 < k ∧ 2 * k ≤ n ∧ (n.choose k).minFac > max (n / k) k}.Finite

number theorybinomial coefficientsN/A2 sources

declared open · formalizederdos:1095

Asymptotics.IsEquivalent Filter.atTop (fun k => Real.log ↑(Erdos1095.g k)) fun k => ↑k / Real.log ↑k

number theorybinomial coefficientsA0034584 sources

declared proved · formalizederdos:1096

True ↔ ∃ ε > 0, ∀ (q : ℝ), 1 < q → q < 1 + ε → ∀ (x : ℕ → ℝ), StrictMono x → Set.range x = {x | ∃ S, ∑ i ∈ S, q ^ i = x} → Filter.Tendsto (fun k => x (k + 1) - x k) Filter.atTop (nhds 0)

number theoryN/A2 sources

declared open · formalizederdos:1097

False ↔ ∃ C > 0, ∀ (A : Finset ℤ), ↑(Erdos1097.CommonDifferencesThreeTermAP A).ncard ≤ C * ↑A.card ^ (3 / 2)

number theoryadditive combinatoricsN/A3 sources

declared proved (Lean) · formalizederdos:1098

True ↔ ∀ (G : Type u_1) [inst : Group G], (∀ (s : Set G), (Erdos1098.nonCommutingGraph G).IsClique s → s.Finite) → ∃ n, ∀ (s : Finset G), (Erdos1098.nonCommutingGraph G).IsClique ↑s → s.card ≤ n

group theoryN/A5 sources

declared open · formalizederdos:1101

Does there exist a good pairwise-coprime sequence

u_n

with

\sum 1/u_n < \infty

and polynomial growth? What if one only requires

u_n \le e^{o(n)}

number theoryN/A4 sources

declared solved (Lean) · formalizederdos:1102

∀ (A : ℕ → ℕ), StrictMono A → Erdos1102.HasPropertyP (Set.range A) → Filter.Tendsto (fun j => ↑(A j) / ↑j) Filter.atTop Filter.atTop

number theoryN/A5 sources

declared open · formalizederdos:1104

∃ c₁, 0 < c₁ ∧ c₁ ≤ 1 ∧ ∀ᶠ (n : ℕ) in Filter.atTop, c₁ * √↑n / √(Real.log ↑n) ≤ ↑(Erdos1104.triangleFreeMaxChromatic n)

graph theorychromatic numberA2925282 sources

declared proved · formalizederdos:1105

True ↔ ∀ (k : ℕ), 3 ≤ k → (fun n => ↑((SimpleGraph.cycleGraph k).antiRamseyNum n) - ((↑k - 2) / 2 + 1 / (↑k - 1)) * ↑n) =O[Filter.atTop] fun x => 1

graph theoryramsey theorypossible3 sources

declared open · formalizederdos:1106

True ↔ Filter.Tendsto (fun n => (∏ i ∈ Finset.Icc 1 n, Erdos1106.p i).primeFactors.card) Filter.atTop Filter.atTop

number theoryA194259 · A1942602 sources

declared open · formalizederdos:1107

∀ r ≥ 2, ∀ᶠ (n : ℕ) in Filter.atTop, Erdos1107.SumOfRPowerful r n

number theorypowerfulA056828 · A392342 · A392343 · possible2 sources

declared open · formalizederdos:1108

True ↔ ∀ k ≥ 2, {a | a ∈ Erdos1108.FactorialSums ∧ ∃ m, m ^ k = a}.Finite

number theoryfactorialsA051761 · A115645 · A0254942 sources

declared open · formalizederdos:1110

True ↔ ∀ (p q : ℕ), q < p → 2 ≤ q → p.Coprime q → ¬(p = 3 ∧ q = 2) → {n | n.Coprime (p * q) ∧ ¬Erdos1110.Representable p q n}.Infinite

number theorypossible2 sources

declared open · formalizederdos:1113

sorry ↔ ∃ k, k.IsSierpinskiNumber ∧ ¬Erdos1113.HasFinitePrimeCoveringSet k

number theorycovering systemsA0763362 sources

declared independent · formalizederdos:1119

True ↔ ∀ (m : Cardinal.{0}), Cardinal.aleph0 < m → m < Cardinal.continuum → ∀ (F : Set (ℂ → ℂ)), (∀ f ∈ F, Differentiable ℂ f) → (∀ (z₀ : ℂ), Cardinal.mk ↑{y | ∃ f ∈ F, f z₀ = y} ≤ m) → Cardinal.mk ↑F ≤ m

analysisset theoryN/A2 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

declared proved (Lean) · formalizederdos:1125

True ↔ ∀ (f : ℝ → ℝ), (∀ (x h : ℝ), h > 0 → 2 * f x ≤ f (x + h) + f (x + 2 * h)) → Monotone f

analysisN/A5 sources

declared proved (Lean) · formalizederdos:1126

True ↔ ∀ (f : ℝ → ℝ), (∀ᵐ (p : ℝ × ℝ) ∂MeasureTheory.volume.prod MeasureTheory.volume, f (p.1 + p.2) = f p.1 + f p.2) → ∃ h, (∀ (x y : ℝ), h (x + y) = h x + h y) ∧ ∀ᵐ (x : ℝ), f x = h x

analysisN/A5 sources

declared disproved · formalized · $50erdos:1128

False ↔ ∀ (A B C : Type), Cardinal.mk A = Cardinal.aleph 1 → Cardinal.mk B = Cardinal.aleph 1 → Cardinal.mk C = Cardinal.aleph 1 → ∀ (f : A → B → C → Fin 2), ∃ A₁ B₁ C₁, Cardinal.mk ↑A₁ = Cardinal.aleph 0 ∧ Cardinal.mk ↑B₁ = Cardinal.aleph 0 ∧ Cardinal.mk ↑C₁ = Cardinal.aleph 0 ∧ Erdos1128.IsMonochromaticBox f A₁ B₁ C₁

set theoryramsey theoryhypergraphsN/A2 sources

declared open · formalizederdos:1133

Must every sufficiently large node set admit bounded labels that force any polynomial fitting almost all labels at degree below

(1+\varepsilon)n

to have arbitrarily large uniform norm? Claimed via Beurling density for Bernstein spaces.

analysispolynomialsN/A4 sources

declared open · formalized · $500erdos:1135

∀ n > 0, ∃ m, CollatzConjecture.collatzStep^[m] n = 1

number theoryA006370 · A0089082 sources

declared proved (Lean) · formalizederdos:1136

True ↔ ∃ A, 1 / 3 < A.lowerDensity ∧ Erdos1136.AvoidsPowersOfTwo A

number theoryN/A5 sources

declared open · formalizederdos:1137

True ↔ Filter.Tendsto (fun x => ↑((Finset.range x).sup fun n => primeGap n * primeGap (n - 1)) / ↑((Finset.range x).sup primeGap) ^ 2) Filter.atTop (nhds 0)

number theoryprimesA083550 · A0052502 sources

declared disproved (Lean) · formalizederdos:1138

False ↔ ∀ C > 1, Asymptotics.IsEquivalent Erdos1138.snd_gt_half_fst (Erdos1138.primeCount_Ioc_mul_const C) fun x => match x with | (x, y) => C * ↑(Erdos1138.sup_primeGap x) / Real.log y

number theoryprimespossible6 sources

declared open · formalizederdos:1139

True ↔ Filter.limsup (fun k => (↑↑(Nat.nth (fun n => 0 < n ∧ ArithmeticFunction.cardFactors n ≤ 2) (k + 1)) - ↑↑(Nat.nth (fun n => 0 < n ∧ ArithmeticFunction.cardFactors n ≤ 2) k)) / ↑(Real.log (↑k + 1))) Filter.atTop = ⊤

number theoryprimesA037143 · A1010413 sources

declared disproved (Lean) · formalizederdos:1141

False ↔ Infinite ↑{n | Erdos1141.Erdos1141Prop n}

number theoryprimesA2145836 sources

declared open · formalizederdos:1142

sorry ↔ Infinite ↑{n | Erdos1142.Erdos1142Prop n}

number theoryprimesA0396692 sources

declared open · formalizederdos:1145

True ↔ Erdos1145.Erdos1145Prop

additive combinatoricsadditive basisN/A2 sources

declared open · formalizederdos:1146

True ↔ Erdos1146.IsEssentialComponent {k | ∃ m n, k = 2 ^ m * 3 ^ n}

number theorypossible2 sources

declared proved (Lean) · formalizederdos:1148

True ↔ ∀ᶠ (n : ℕ) in Filter.atTop, Erdos1148.Erdos1148Prop n

number theoryA390380 · A3931686 sources

declared open · formalizederdos:1150

True ↔ ∃ c > 0, ∀ᶠ (n : ℕ) in Filter.atTop, ∀ (P : Polynomial ℂ), (∀ i ≤ P.natDegree, P.coeff i = -1 ∨ P.coeff i = 1) → P.natDegree = n → ⨆ z, ‖Polynomial.eval (↑z) P‖ > (1 + c) * √↑n

analysispolynomialsN/A2 sources

declared open · formalizederdos:1167

∀ (μ : Cardinal.{u}) (r : ℕ) (ν : Ordinal.ToType 1 → Cardinal.{u}), Combinatorics.cardinalPartitionRel μ r 1 ν ↔ μ ≥ ν Erdos1167.i0

set theoryprobabilityN/A2 sources

declared open · formalizederdos:1175

True ↔ ∀ (κ : Cardinal.{u_1}), Cardinal.aleph0 < κ → ∃ μ, ∀ (V : Type u_1) (G : SimpleGraph V), G.chromaticCardinal = μ → ∃ H, H.coe.CliqueFree 3 ∧ H.coe.chromaticCardinal = κ

set theorychromatic numberN/A2 sources

declared not disprovable · formalizederdos:1176

True ↔ ∀ {V : Type u_1} (G : SimpleGraph V), G.chromaticCardinal = Cardinal.aleph 1 → ∃ EColor, ∃ (_ : Cardinal.mk EColor = Cardinal.aleph 1), ∃ c_edge, ∀ (VColor : Type), Cardinal.mk VColor ≤ Cardinal.aleph 0 → ∀ (c_vert : V → VColor), ∃ vc, ∀ (ec : EColor), ∃ u v, ∃ (h : G.Adj u v), c_vert u = vc ∧ c_vert v = vc ∧ c_edge ⟨s(u, v), h⟩ = ec

set theorychromatic numberN/A2 sources

declared open · formalizederdos:1188

Estimate the number

F(x)

of minimal distinct covering systems whose moduli all lie in

[1, x]

. The candidate proof gives

\log\log F(x)/\log x \to 1

, i.e.

F(x) = \exp(x^{1+o(1)})

number theorycovering systemspossible4 sources

declared solved (Lean) · formalizederdos:1190

∀ (ε : ℝ), 0 < ε → ∀ᶠ (m : ℕ) in Filter.atTop, scaleL m ^ (-1 - ε) < Erdos1190.eps m ∧ Erdos1190.eps m < scaleL m ^ (-1 + ε)

number theorycovering systemspossible6 sources

declared open · formalizederdos:1192

True ↔ ∀ r ≥ 2, ∃ A, (∀ᶠ (n : ℕ) in Filter.atTop, Erdos1192.f_r A r n > 0) ∧ (fun x => ∑ n ∈ Finset.range (x + 1), ↑(Erdos1192.f_r A r n) ^ 2) =O[Filter.atTop] fun x => ↑x

additive combinatoricsadditive basispossible2 sources

declared solved (Lean) · formalizederdos:1193

False ↔ ∀ (A : Set ℕ) (g : ℕ → ℕ), Monotone g → (∀ (n : ℕ), 0 < g n) → {n | AdditiveCombinatorics.sumRep A n = g n}.lowerDensity = 0

additive combinatoricsN/A5 sources

declared proved (Lean) · formalizederdos:1196

Bounds the weighted sum

\sum 1/(a \log a)

taken over primitive sets of integers (sets where no element divides another).

number theoryprimitive setspossible6 sources

declared open · formalizederdos:1199

True ↔ ∀ (color : ℕ → Fin 2), ∃ A, A.Infinite ∧ ∀ n ∈ A + A, ∀ m ∈ A + A, color n = color m

additive combinatoricsramsey theorypossible2 sources

declared open · formalizederdos:1201

Is it true that for every

\epsilon,\eta>0

there exists a

k

such that the density of

n

for which

P(n(n+1)\cdots(n+k))>n^{1-\epsilon}

is at least

1-\eta

, where

P(m)

is the greatest prime divisor of

m

? A short argument via the Matomäki-Radziwiłł theorem establishes the lower-density version.

number theoryprimespossible4 sources

declared open · formalizederdos:1203

True ↔ Filter.Tendsto Erdos1203.F Filter.atTop Filter.atTop

number theorypossible2 sources

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