Vela Mathematics Program: problems

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1,217 problems · 24/25Number SourcesGraph view

declared openerdos:1151

\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 provederdos:1153

Erdős problem 1153

analysispolynomialsN/A3 sources

declared not disprovableerdos:1154

Erdős problem 1154

analysisN/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 solvederdos:1161

Erdős problem 1161

group theorypossible

declared openerdos:1162

Erdős problem 1162

group theorypossible

declared openerdos:1163

Erdős problem 1163

group theoryN/A

declared provederdos:1164

Erdős problem 1164

probabilityN/A

declared solvederdos:1165

Erdős problem 1165

probabilityN/A

declared provederdos:1166

Erdős problem 1166

probabilitypossible

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 not disprovableerdos:1169

Erdős problem 1169

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 not disprovableerdos:1174

Erdős problem 1174

set theoryramsey theoryN/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 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 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 provederdos:1179

Erdős problem 1179

additive combinatoricsprobabilityN/A

declared provederdos:1180

Erdős problem 1180

number theoryN/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 solvederdos:1185

Erdős problem 1185

additive combinatoricsarithmetic progressionsN/A

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 solvederdos:1187

Erdős problem 1187

number theoryprimesadditive combinatoricsN/A3 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 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 · $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 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 disprovederdos:1198

Erdős problem 1198

additive combinatoricsramsey theoryN/A

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

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