Vela Mathematics Program: problems

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604 problems · 12/13Number SourcesGraph view

declared openerdos:1120

Erdős problem 1120

analysisN/A

declared openerdos:1122

Erdős problem 1122

number theoryN/A

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 open · formalized · $500erdos:1135

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

number theoryA006370 · A0089082 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 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 open · formalizederdos:1142

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

number theoryprimesA0396692 sources

declared openerdos:1143

Erdős problem 1143

number theoryprimesN/A2 sources

declared openerdos:1144

Erdős problem 1144

number theoryprobabilityN/A

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 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 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 openerdos:1152

Erdős problem 1152

analysispolynomialsN/A

declared openerdos:1155

Erdős problem 1155

graph theoryN/A

declared openerdos:1156

Erdős problem 1156

graph theorychromatic numberN/A

declared openerdos:1157

Erdős problem 1157

hypergraphsturan numberpossible

declared openerdos:1158

Erdős problem 1158

hypergraphsturan numberpossible

declared openerdos:1159

Erdős problem 1159

combinatoricsN/A

declared openerdos:1160

Erdős problem 1160

group theoryA000001

declared openerdos:1162

Erdős problem 1162

group theorypossible

declared openerdos:1163

Erdős problem 1163

group theoryN/A

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 openerdos:1168

Erdős problem 1168

set theoryramsey theoryN/A

declared openerdos:1170

Erdős problem 1170

set theoryramsey theoryN/A

declared openerdos:1171

Erdős problem 1171

set theoryramsey theoryN/A

declared openerdos:1172

Erdős problem 1172

set theoryramsey theoryN/A

declared openerdos:1173

Erdős problem 1173

set theorycombinatoricsN/A

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 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:1178

Erdős problem 1178

graph theoryhypergraphsN/A

declared openerdos:1181

Erdős problem 1181

number theorypossible

declared openerdos:1182

Erdős problem 1182

graph theoryramsey theorypossible

declared openerdos:1183

Erdős problem 1183

combinatoricsramsey theorypossible2 sources

declared openerdos:1184

Erdős problem 1184

number theoryprimespossible

declared openerdos:1186

What is the minimum asymptotic density

\delta_k

of monochromatic

k

-term arithmetic progressions in every two-colouring of

\{1, \dots, n\}

? The exact certificate gives

\delta_3 = 117/2192

, matching the known 548-bead colouring.

additive combinatoricsarithmetic progressionspossible2 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 openerdos:1189

For irreducible covering sets of size

k

, determine their count, the possible largest modulus, the maximal reciprocal sum, and whether divisor-set examples occur infinitely often.

number theorycovering systemspossible2 sources

declared open · $1000erdos:1191

Erdős problem 1191

additive combinatoricssidon setspossible

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 openerdos:1194

Erdős problem 1194

additive combinatoricsadditive basissidon setspossible2 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 openerdos:1200

Erdős problem 1200

number theoryprimespossible

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

declared openerdos:1204

Erdős problem 1204

number theoryA008407 · A023193 · A135311 · possible

declared openerdos:1206

Erdős problem 1206

number theorysidon setspossible

declared openerdos:1207

Erdős problem 1207

geometrydistancespossible

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