Vela Mathematics Program: problems

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319 problems · 4/7Number SourcesGraph view

declared open · formalizederdos:455

True ↔ ∀ (q : ℕ → ℕ), StrictMono q → (∀ (n : ℕ), Nat.Prime (q n) ∧ q (n + 2) - q (n + 1) ≥ q (n + 1) - q n) → Filter.Tendsto (fun n => ↑(q n) / ↑n ^ 2) Filter.atTop Filter.atTop

number theoryN/A2 sources

declared open · formalizederdos:456

True ↔ Filter.Tendsto (fun N => ↑(Nat.count {n | Erdos456.m n < Erdos456.p n} N) / ↑N) Filter.atTop (nhds 1)

number theoryA034694 · A061026 · possible3 sources

declared open · formalizederdos:463

True ↔ ∃ f, ∃ (_ : Filter.Tendsto f Filter.atTop Filter.atTop), ∀ᶠ (n : ℕ) in Filter.atTop, ∃ m, m.Composite ∧ n + f n < m ∧ m < n + m.minFac

number theoryprimespossible2 sources

declared open · formalized · $10erdos:470

sorry ↔ ∃ n, n.Weird ∧ Odd n

number theorydivisorsA006037 · A0029752 sources

declared open · formalizederdos:477

Does there exist an integer polynomial

f

of degree at least two and a set

A \subseteq \mathbb{Z}

such that every integer has a unique representation

n = a + f(k)

? A manuscript claims the thirteenth powers admit a tiling complement.

number theoryN/A4 sources

declared open · formalizederdos:479

sorry ↔ ∀ k > 1, {n | 2 ^ n ≡ k [MOD n]}.Infinite

number theoryA036236 · A015919 · A050259 · A015921 · A006521 · A006517 · A0159402 sources

declared open · formalizederdos:486

True ↔ ∀ (X : (n : ℕ) → Set (ZMod n)), ∃ d, {m | ∀ (n : ℕ), ↑m ∉ X n}.HasLogDensity d

number theoryprimitive setsN/A2 sources

declared open · formalizederdos:489

A

is a forbidden-divisor set with

|A \cap [1,x]| = o(\sqrt{x})

and

B = \{b_1 < b_2 < \cdots\}

the sifted set, must

x^{-1} \sum_{b_i < x} (b_{i+1} - b_i)^2

converge to a finite limit?

number theoryN/A4 sources

declared open · formalizederdos:495

True ↔ ∀ (α β : ℝ), Filter.liminf (fun n => ↑n * distToNearestInt (↑n * α) * distToNearestInt (↑n * β)) Filter.atTop = 0

diophantine approximationnumber theoryN/A2 sources

declared open · formalizederdos:501

For every

x \in \mathbb{R}

let

A_x \subset \mathbb{R}

be a bounded set of Lebesgue outer measure

< 1

. Must there be an infinite independent set, that is an infinite

X \subseteq \mathbb{R}

with

x \notin A_y

for all distinct

x, y \in X

? Erdős and Hajnal proved that arbitrarily large finite independent sets exist. Hechler showed in 1972 that the answer is no under the continuum hypothesis, so any positive answer had to come from a model where CH fails, and Sungchul Lee later derived one from a real-valued measurable cardinal. The answer is that neither side is provable. Dropping Lee's large cardinal by transferring his argument to the extension of a model of CH by random reals gives a model where the answer is yes; Hechler's construction gives one where it is no. The question is independent of ZFC.

combinatoricsset theoryN/A4 sources

declared open · formalizederdos:503

∀ (n : ℕ), IsGreatest {x | ∃ A, ∃ (_ : A.IsIsosceles), A.ncard = x} sorry

geometrydistancesA1757693 sources

declared open · formalizederdos:507

Asymptotics.IsEquivalent Filter.atTop Erdos507.α sorry

geometryN/A3 sources

declared open · formalizederdos:508

4 ≤ (SimpleGraph.UnitDistancePlaneGraph Set.univ).chromaticNumber

geometryramsey theoryN/A2 sources

declared open · formalizederdos:509

∀ {M : Type u} [inst : MetricSpace M] (S : Set M), S.Nonempty → ∀ (r : ℝ) (ι : Type v) (bdc : Erdos509.BoundedDiscCover S r ι), 0 < r

analysisN/A2 sources

declared open · formalizederdos:510

sorry ↔ ∃ c, ∃ (_ : 0 < c), ∀ᶠ (N : ℕ) in Filter.atTop, ∀ (A : Finset ℕ), 0 ∉ A → A.card = N → ∃ θ, ∑ n ∈ A, Real.cos (↑n * θ) < -c * √↑N

analysisN/A2 sources

declared open · formalizederdos:513

sorry = ⨆ f, Filter.liminf (fun r => Erdos513.ratio r ↑f) Filter.atTop

analysisN/A3 sources

declared open · formalizederdos:517

sorry ↔ ∀ {f : ℂ → ℂ} {n : ℕ → ℕ}, HasFabryGaps n → ∀ {a : ℕ → ℂ}, (∀ (k : ℕ), a k ≠ 0) → (∀ (z : ℂ), HasSum (fun k => a k * z ^ n k) (f z)) → ∀ (z : ℂ), {x | f x = z}.Infinite

analysisN/A3 sources

declared open · formalizederdos:520

True ↔ ∃ c > 0, ∀ (Ω : Type) [inst : MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (f : ℕ → Ω → ℝ), Erdos520.IsRademacherMultiplicative f → ∀ᵐ (ω : Ω), Filter.limsup (fun N => ∑ m ∈ Finset.Iic N, f m ω / √(↑N * Real.log (Real.log ↑N))) Filter.atTop = c

number theoryprobabilityN/A2 sources

declared open · formalizederdos:521

False ↔ Erdos521.Claim

analysispolynomialsprobabilityN/A4 sources

declared open · formalizederdos:522

For

P_n(z) = \sum_{k=0}^n \varepsilon_k z^k

with independent uniform signs, does the number

R_n

of roots in

|z| \le 1

satisfy

R_n/(n/2) \to 1

almost surely? The manuscript proves the strong law with

R_n = n/2 + O_\omega(n^{149/150})

analysispolynomialsprobabilityN/A4 sources

declared open · formalizederdos:535

∀ r ≥ 3, ∃ c > 0, ∀ᶠ (N : ℕ) in Filter.atTop, ↑(Erdos535.f r N) ≤ ↑N ^ (c / Real.log (Real.log ↑N))

number theorypossible3 sources

declared open · formalizederdos:536

True ↔ ∀ ε > 0, ∀ᶠ (N : ℕ) in Filter.atTop, ∀ A ⊆ Finset.Icc 1 N, ε * ↑N ≤ ↑A.card → ∃ a ∈ A, ∃ b ∈ A, ∃ c ∈ A, {a, b, c}.card = 3 ∧ a.lcm b = b.lcm c ∧ b.lcm c = a.lcm c

number theorypossible3 sources

declared open · formalizederdos:538

If each integer has at most

r

representations

m = pa

with

p

prime and

a \in A \subseteq [1, N]

, what is the best upper bound for

\sum_{a \in A} 1/a

? The candidate proof gives the matching order

\Theta_r(\log N / \log\log N)

number theoryN/A5 sources

declared open · formalizederdos:539

For

|A| = n

, how small can the cofactor set

Q(A) = \{a / \gcd(a,b) : a, b \in A\}

be? The answer is

h(n) = n^{1/2 + o(1)}

: a new upper bound

h(n) \le n^{1/2} \exp(O(\sqrt{\log n}))

matches the classical lower bound.

number theorypossible5 sources

declared open · formalizederdos:562

True ↔ ∀ r ≥ 3, (fun n => Real.log^[r - 1] ↑(Combinatorics.hypergraphRamsey r n)) =Θ[Filter.atTop] fun n => ↑n

graph theoryramsey theoryhypergraphspossible2 sources

declared open · formalized · $500erdos:564

True ↔ ∃ c > 0, ∀ᶠ (n : ℕ) in Filter.atTop, 2 ^ 2 ^ (c * n) ≤ ↑(Combinatorics.hypergraphRamsey 3 n)

graph theoryramsey theoryhypergraphspossible2 sources

declared open · formalizederdos:566

True ↔ ∀ (p : ℕ) (G : SimpleGraph (Fin p)), (∀ (S : Finset (Fin p)), 2 ≤ S.card → (SimpleGraph.induce (↑S) G).edgeSet.ncard ≤ 2 * S.card - 3) → ∃ c > 0, ∀ (n : ℕ) (H : SimpleGraph (Fin n)) [inst : DecidableRel H.Adj], (∀ (v : Fin n), 0 < H.degree v) → ↑(G.sizeRamsey H) ≤ c * ↑H.edgeSet.ncard

graph theoryramsey theoryN/A2 sources

declared open · formalizederdos:567

True ↔ Erdos567.Q3.IsRamseySizeLinear

graph theoryramsey theoryN/A2 sources

declared open · formalizederdos:579

True ↔ ∀ (δ : ℝ), 0 < δ → ∃ c, 0 < c ∧ ∀ᶠ (n : ℕ) in Filter.atTop, ∀ (G : SimpleGraph (Fin n)), Erdos579.octahedron.Free G → δ * ↑n ^ 2 ≤ ↑G.edgeFinset.card → c * ↑n ≤ ↑G.indepNum

graph theoryturan numberN/A2 sources

declared open · formalized · $1000erdos:592

∀ (β : Ordinal.{u}), β.card ≤ Cardinal.aleph0 → OrdinalCardinalRamsey (Ordinal.omega0 ^ β) (Ordinal.omega0 ^ β) 3 ↔ sorry β

set theoryramsey theoryN/A2 sources

declared open · formalized · $500erdos:593

Erdos Problem #593 asks which finite triple systems occur in every uncountably chromatic triple system. The answer is exactly the class generated from private-vertex expansions of finite bipartite graphs by finite disjoint unions and one-point amalgamations. Problem #1177 on exact spectra is settled alongside it.

set theorygraph theoryhypergraphsN/A4 sources

declared open · formalized · $250erdos:595

True ↔ ∃ V, ∃ (_ : Infinite V), ∃ G, G.CliqueFree 4 ∧ ¬Erdos595.IsCountableUnionOfTriangleFree G

graph theoryset theoryN/A2 sources

declared open · formalizederdos:596

∀ {U₁ U₂ : Type} (G₁ : SimpleGraph U₁) (G₂ : SimpleGraph U₂), G₁.IsErdosHajnalExceptional G₂ ↔ (fun {U₁ U₂} => sorry) G₁ G₂

graph theoryramsey theoryset theoryN/A2 sources

declared open · formalizederdos:598

∀ (m : Type u_1) [Infinite m], True ↔ ∃ c, ∀ (X : Set m), Cardinal.mk ↑X = Erdos598.κ → c '' {s | ↑s ⊆ X} = Set.univ

set theoryramsey theoryN/A3 sources

declared open · formalizederdos:600

True ↔ ∀ (r : ℕ), 2 ≤ r → Filter.Tendsto (fun n => ↑(Erdos600.eFunction n (r + 1)) - ↑(Erdos600.eFunction n r)) Filter.atTop Filter.atTop

graph theorypossible2 sources

declared open · formalizederdos:602

¬Erdos602.disjoint_without_infinite_claim

combinatoricsset theoryN/A3 sources

declared open · formalizederdos:623

Let

X

be a set of cardinality

\aleph_\omega

and

f

a function from the finite subsets of

X

X

such that

f(A)\not\in A

for all

A

. Must there exist an infinite independent

Y\subseteq X

, i.e. with

f(B)\not\in Y

for all finite

B\subset Y

? Claimed resolution: the positive assertion is equivalent to Koepke's free-subset property, hence independent of ZFC, with consistency strength exactly a measurable cardinal.

set theoryN/A4 sources

declared open · formalizederdos:624

Filter.Tendsto (fun n => ↑(Erdos624.H n) - Real.logb 2 ↑n) Filter.atTop Filter.atTop

combinatoricspossible2 sources

declared open · formalizederdos:653

True ↔ ∃ o, o =o[Filter.atTop] 1 ∧ ∀ᶠ (n : ℕ) in Filter.atTop, (1 - o n) * ↑n ≤ ↑(EuclideanGeometry.maximalDistinctDistancesFrom n)

geometrydistancespossible2 sources

declared open · formalizederdos:655

False ↔ ∃ c > 0, ∀ᶠ (n : ℕ) in Filter.atTop, ∀ (X : Finset (EuclideanSpace ℝ (Fin 2))), X.card = n → Erdos655.IsValid X → (1 + c) * ↑n / 2 ≤ ↑(EuclideanGeometry.distinctDistances X)

geometrydistancespossible3 sources

declared open · formalizederdos:677

∀ (m n k : ℕ), k > 0 → m ≥ n + k → Finset.lcmInterval m k ≠ Finset.lcmInterval n k

number theorypossible2 sources

declared open · formalizederdos:680

True ↔ ∀ᶠ (n : ℕ) in Filter.atTop, ∃ k, k ≠ 0 ∧ (n + k).minFac > k ^ 2 + 1

number theoryprimesN/A2 sources

declared open · formalizederdos:681

True ↔ ∀ᶠ (n : ℕ) in Filter.atTop, ∃ k > 0, (n + k).Composite ∧ ∀ (p : ℕ), Erdos681.IsLPF p (n + k) → p > k ^ 2

number theoryprimesA3896802 sources

declared open · formalizederdos:683

True ↔ ∃ c > 0, ∀ (n k : ℕ), 0 < k ∧ k < n → ↑(Erdos683.P n k) > min (↑n - ↑k + 1) (↑k ^ (1 + c))

number theoryprimesbinomial coefficientsA006530 · A074399 · A121359 · possible2 sources

declared open · formalizederdos:686

sorry ↔ ∀ N ≥ 2, ∃ k ≥ 2, ∃ n, ∃ m ≥ n + k, ↑N = ↑(∏ i ∈ Finset.Icc 1 k, (m + i)) / ↑(∏ i ∈ Finset.Icc 1 k, (n + i))

number theoryN/A4 sources

declared open · formalizederdos:688

sorry =O[Filter.atTop] Erdos688.epsilonFunction

number theoryN/A2 sources

declared open · formalizederdos:689

sorry ↔ ∀ᶠ (n : ℕ) in Filter.atTop, ∃ a, ∀ m ∈ Finset.Icc 1 n, 2 ≤ {p ∈ Finset.Icc 1 n | Nat.Prime p ∧ a p ≡ m [MOD p]}.card

number theoryN/A3 sources

declared open · formalizederdos:695

sorry ↔ ∀ {q : ℕ → ℕ}, StrictMono q → (∀ (i : ℕ), Nat.Prime (q i)) → (∀ (i : ℕ), q (i + 1) % q i = 1) → Filter.Tendsto (fun k => ↑(q k) ^ (1 / ↑k)) Filter.atTop Filter.atTop

number theoryA0610922 sources

declared open · formalizederdos:700

∀ (n : ℕ), ¬Nat.Prime n → 1 < n → (Erdos700.f n = n / Erdos700.P n ↔ True)

number theorybinomial coefficientsA091963 · possible2 sources

declared open · formalizederdos:701

True ↔ ∀ {X : Type} [Nonempty X] [Fintype X] (F : Set (Set X)), IsLowerSet F → ∃ x, ∀ F' ⊆ F, F'.Intersecting → Cardinal.mk ↑F' ≤ Cardinal.mk ↑{A | A ∈ F ∧ x ∈ A}

combinatoricsintersecting familyN/A2 sources

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