Vela Mathematics Program: problems

Problem ledger

1,217 problems · 21/25Number SourcesGraph view

declared solvederdos:1001

number theorydiophantine approximationN/A

declared open · formalizederdos:1002

sorry ↔ ∃ g, Monotone g ∧ Filter.Tendsto g Filter.atBot (nhds 0) ∧ Filter.Tendsto g Filter.atTop (nhds 1) ∧ ∀ (c : ℝ), Filter.Tendsto (fun n => (MeasureTheory.volume {α | α ∈ Set.Ioo 0 1 ∧ (fun α n => 1 / Real.log ↑n * ∑ k ∈ Finset.Icc 1 n, (1 / 2 - Int.fract (α * ↑k))) α n ≤ c}).toReal) Filter.atTop (nhds (g c))

analysisdiophantine approximationN/A2 sources

declared open · formalizederdos:1003

True ↔ {n | n.totient = (n + 1).totient}.Infinite

number theoryA0012742 sources

declared open · formalizederdos:1004

True ↔ ∀ c > 0, ∀ᶠ (x : ℕ) in Filter.atTop, ∃ n ≤ x, Erdos1004.IsDistinctTotientRun n ⌊Real.log ↑x ^ c⌋₊

number theorypossible3 sources

declared openerdos:1005

Erdős problem 1005

number theoryA386893

declared disprovederdos:1006

Erdős problem 1006

graph theorycyclesN/A

declared solved (Lean) · formalizederdos:1007

IsLeast {m | ∃ n G, G.HasDimension 4 ∧ G.edgeSet.ncard = m} sorry

graph theoryN/A5 sources

declared proved (Lean) · formalizederdos:1008

True ↔ ∃ c > 0, ∀ (V : Type) [Fintype V] (G : SimpleGraph V), ∃ H ≤ G, (SimpleGraph.cycleGraph 4).Free H ∧ c * ↑G.edgeSet.ncard ^ (2 / 3) ≤ ↑H.edgeSet.ncard

graph theoryN/A5 sources

declared provederdos:1009

Erdős problem 1009

graph theoryN/A

declared provederdos:1010

Erdős problem 1010

graph theoryN/A

declared openerdos:1011

Erdős problem 1011

graph theorypossible

declared solvederdos:1012

Erdős problem 1012

graph theory

declared openerdos:1013

Erdős problem 1013

graph theoryA292528

declared proved (Lean) · formalizederdos:1014

∀ (k : ℕ), 3 ≤ k → Filter.Tendsto (fun l => ↑R(k, l + 1) / ↑R(k, l)) Filter.atTop (nhds 1)

graph theorypossible7 sources

declared solvederdos:1015

Erdős problem 1015

graph theoryramsey theorypossible

declared openerdos:1016

Erdős problem 1016

graph theorycyclesA105206

declared openerdos:1017

Erdős problem 1017

graph theorypossible

declared solvederdos:1018

Erdős problem 1018

graph theoryplanar graphsN/A

declared provederdos:1019

Erdős problem 1019

graph theoryplanar graphsN/A

declared falsifiableerdos:1020

Erdős problem 1020

graph theoryhypergraphsN/A

declared provederdos:1021

Erdős problem 1021

graph theoryN/A2 sources

declared proved (Lean) · formalizederdos:1022

False ↔ ∃ c, Filter.Tendsto c Filter.atTop Filter.atTop ∧ ∀ (t : ℕ), Erdos1022.SparseImpliesPropertyB t (c t)

combinatoricshypergraphspossible6 sources

declared solved (Lean) · formalizederdos:1023

True ↔ ∃ c, 0 < c ∧ Asymptotics.IsEquivalent Filter.atTop (fun n => ↑(Erdos1023.F n)) fun n => c * 2 ^ n / ↑n ^ (1 / 2)

combinatoricsA347025 · possible5 sources

declared solvederdos:1024

Erdős problem 1024

graph theoryhypergraphspossible

declared solvederdos:1025

Erdős problem 1025

combinatoricspossible

declared solved (Lean) · formalizederdos:1026

For a sequence of

n

distinct reals, determine the largest constant

c

such that some monotonic subsequence always has sum exceeding

(c-o(1))\cdot(1/\sqrt{n})

times the total sum. Resolved as

c = 1

combinatoricsA391431 · A3914906 sources

declared provederdos:1027

Erdős problem 1027

combinatoricsN/A

declared solved (Lean) · formalizederdos:1028

(fun n => ↑(Erdos1028.H n)) =Θ[Filter.atTop] fun n => ↑n ^ (3 / 2)

graph theorydiscrepancypossible5 sources

declared open · $100erdos:1029

Erdős problem 1029

graph theoryramsey theoryA059442

declared openerdos:1030

Erdős problem 1030

graph theoryramsey theoryA000791 · A059442

declared provederdos:1031

Erdős problem 1031

graph theorypossible

declared openerdos:1032

Do arbitrarily large 4-chromatic edge-critical graphs exist with minimum degree bounded below by a positive constant times the number of vertices?

graph theorychromatic numberpossible3 sources

declared openerdos:1033

Erdős problem 1033

graph theorypossible2 sources

declared disproved (Lean) · formalizederdos:1034

False ↔ ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in Filter.atTop, ∀ (G : SimpleGraph (Fin n)), ↑n ^ 2 / 4 < ↑G.edgeSet.ncard → ∃ T, G.IsNClique 3 T ∧ ∃ Y, Erdos1034.JoinedToTwo G T Y ∧ (1 / 2 - ε) * ↑n < ↑Y.card

graph theorypossible5 sources

declared openerdos:1035

Erdős problem 1035

graph theorypossible

declared proved (Lean) · formalizederdos:1036

True ↔ ∀ (c : ℝ), 0 < c → ∃ δ, 0 < δ ∧ ∀ᶠ (n : ℕ) in Filter.atTop, ∀ (G : SimpleGraph (Fin n)), ↑G.cliqueNum ≤ c * Real.log ↑n → ↑G.indepNum ≤ c * Real.log ↑n → Erdos1036.HasManyNonIsomorphicInducedSubgraphs G (2 ^ (δ * ↑n))

graph theorypossible5 sources

declared disproved (Lean) · formalizederdos:1037

False ↔ ∀ (ε : ℝ), 0 < ε → ∀ (C : ℝ), ∀ᶠ (n : ℕ) in Filter.atTop, ∀ (G : SimpleGraph (Fin n)), (∀ (d : ℕ), {v | (G.neighborSet v).ncard = d}.ncard ≤ 2) → (1 / 2 + ε) * ↑n < ↑(Set.range fun v => (G.neighborSet v).ncard).ncard → ∃ s, Erdos1037.IsTrivialSet G s ∧ C * Real.log ↑n < ↑s.ncard

graph theoryN/A5 sources

declared open · formalizederdos:1038

Among all nonconstant monic polynomials

f

whose roots lie in

[-1, 1]

, determine

\inf_f |\{x \in \mathbb{R} : |f(x)| < 1\}|

analysisN/A4 sources

declared openerdos:1039

For

f(z) = \prod_{i=1}^n (z - z_i)

with all

|z_i| \le 1

, let

\rho(f)

be the radius of the largest disc contained in

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

. Is

\rho(f) \gg 1/n

? The worst case is now known to be

\Theta(1/n)

, with the explicit bound

\rho(f) \ge (\log 2)/n

analysisN/A3 sources

declared openerdos:1040

For a closed infinite set

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 provederdos:1042

Erdős problem 1042

analysisN/A

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 openerdos:1045

Erdős problem 1045

analysisN/A

declared disprovederdos:1046

Erdős problem 1046

analysisN/A

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 open · formalizederdos:1049

True ↔ ∀ t > 1, Irrational (∑' (n : ℕ+), 1 / (↑t ^ ↑n - 1))

irrationalityN/A2 sources

declared provederdos:1050

Erdős problem 1050

irrationalityA331372

Problems · Vela Mathematics Program

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