Vela Mathematics Program: problems

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353 problems · 7/8Number SourcesGraph view

declared openerdos:1040

F \subseteq \mathbb{C}

, let

\mu(F)

be the infimum of

|\{z : |f(z)| < 1\}|

over monic polynomials with zeros in

F

. Is

\mu(F)

determined only by the transfinite diameter of

F

analysisN/A3 sources

declared falsifiable · formalizederdos:1041

∀ (n : ℕ) (f : Polynomial ℂ), n ≥ 2 → f.natDegree = n → f.Monic → f.rootSet ℂ ⊆ Metric.ball 0 1 → ∃ z₁ z₂, ∃ (_ : {z₁, z₂} ≤ f.roots), ∃ γ, Set.range ⇑γ ⊆ {z | ‖Polynomial.eval z f‖ < 1} ∧ Erdos1041.length (Set.range ⇑γ) < 2

analysisN/A3 sources

declared disproved (Lean) · formalizederdos:1043

False ↔ ∀ (f : Polynomial ℂ), f.Monic → f.degree ≥ 1 → ∃ u, ‖u‖ = 1 ∧ MeasureTheory.volume (⇑(ℝ ∙ u).orthogonalProjection '' Erdos1043.levelSet f) ≤ 2

analysisN/A5 sources

declared solved (Lean) · formalizederdos:1044

IsGLB {L | ∃ f, Erdos1044.IsAdmissible f ∧ Erdos1044.maxBoundaryLength f = L} 2

analysisN/A5 sources

declared disproved (Lean) · formalizederdos:1047

False ↔ ∀ (f : Polynomial ℂ) (m : ℕ) (c : ℝ), f.Monic → (f.rootSet ℂ).ncard = m → 0 < c → (Erdos1047.componentsIn (Erdos1047.sublevelSet f c)).ncard = m → ∀ t ∈ Erdos1047.componentsIn (Erdos1047.sublevelSet f c), Convex ℝ t

analysisN/A5 sources

declared disproved (Lean) · formalizederdos:1048

False ↔ ∀ (r : ℝ) (f : Polynomial ℂ), r < 2 → f.Monic → f.degree ≥ 1 → (∀ z ∈ f.roots, ‖z‖ ≤ r) → ∃ z ∈ Erdos1048.openLevelSet f, ENNReal.ofReal (2 - r) < Metric.ediam (connectedComponentIn (Erdos1048.openLevelSet f) z)

analysisN/A5 sources

declared proved (Lean) · formalizederdos:1051

True ↔ ∀ (a : ℕ → ℤ), StrictMono a → Erdos1051.GrowthCondition a → Irrational (Erdos1051.ErdosSeries a)

irrationalityN/A6 sources

declared open · formalizederdos:1054

sorry ↔ (fun n => ↑(Erdos1054.f n)) =o[Filter.atTop] fun n => ↑n

number theorydivisorsA1674853 sources

declared open · formalizederdos:1061

For

S(x) = \#\{(a,b) : a + b \le x,\ \sigma(a) + \sigma(b) = \sigma(a+b)\}

, is

S(x) \sim cx

? The preprint claims

S(x)

grows faster than

x (\log x)^R

for every fixed

R

, ruling out the linear asymptotic.

number theoryA110177 · possible4 sources

declared open · formalizederdos:1062

(∃ l, Filter.Tendsto (fun n => ↑(Erdos1062.f n) / ↑n) Filter.atTop (nhds l) ∧ Irrational l) ↔ True

number theoryA0383723 sources

declared disproved (Lean) · formalizederdos:1067

False ↔ ∀ (V : Type) (G : SimpleGraph V), G.chromaticCardinal = Cardinal.aleph 1 → ∃ H, H.coe.chromaticCardinal = Cardinal.aleph 1 ∧ H.coe.InfinitelyConnected

graph theoryset theoryN/A5 sources

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 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 solvederdos:1089

Erdős problem 1089

geometrydistancespossible3 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 solvederdos:1091

Erdős problem 1091

geometrychromatic numberN/A3 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:1095

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

number theorybinomial coefficientsA0034584 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 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 openerdos:1131

Erdős problem 1131

analysispolynomialsN/A2 sources

declared openerdos:1132

Erdős problem 1132

analysispolynomialsN/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 disproved (Lean) · £10erdos:1134

Erdős problem 1134

number theoryA1856613 sources

declared proved (Lean) · formalizederdos:1136

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

number theoryN/A5 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 openerdos:1143

Erdős problem 1143

number theoryprimesN/A2 sources

declared proved (Lean) · formalizederdos:1148

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

number theoryA390380 · A3931686 sources

declared openerdos:1151

Let

\mathcal{L}^nf

be the Lagrange interpolation polynomials of a continuous

f

on the Chebyshev nodes. Prove that, for any closed

A\subseteq [-1,1]

, there exists a continuous function

f

such that

A

is the set of limit points of

\mathcal{L}^nf(x)

analysispolynomialsN/A3 sources

declared provederdos:1153

Erdős problem 1153

analysispolynomialsN/A3 sources

declared openerdos:1177

For a finite forbidden triple system

G

, what exact uncountable chromatic cardinalities occur among

G

-free triple systems, and how do those spectra interact? The revised manuscript answers the three exact-cardinal questions and claims a complete spectrum dichotomy.

set theorychromatic numberhypergraphsN/A3 sources

declared openerdos:1183

Erdős problem 1183

combinatoricsramsey theorypossible2 sources

declared solvederdos:1187

Erdős problem 1187

number theoryprimesadditive combinatoricsN/A3 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 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 openerdos:1194

Erdős problem 1194

additive combinatoricsadditive basissidon setspossible2 sources

declared solvederdos:1195

Erdős problem 1195

number theorypossible3 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 disproved (Lean)erdos:1197

Erdős problem 1197

analysisN/A5 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 solvederdos:1202

Erdős problem 1202

number theoryprimespossible3 sources

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