Vela Mathematics Program: problems

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111 problems · 2/3Number SourcesGraph view

declared proved (Lean) · formalizederdos:443

True ↔ ∀ (s : ℕ), ∃ m, ∃ n < m, s ≤ (Erdos443.A n ∩ Erdos443.A m).card

number theoryN/A5 sources

declared proved (Lean) · formalizederdos:447

True ↔ (fun n => ↑(Erdos447.maxUnionFree n)) =o[Filter.atTop] fun n => 2 ^ n

combinatoricspossible5 sources

declared proved (Lean) · formalizederdos:457

True ↔ ∃ ε > 0, {n | ∀ (p : ℕ), ↑p ≤ (2 + ε) * Real.log ↑n → Nat.Prime p → p ∣ ∏ i ∈ Finset.Icc 1 ⌊Real.log ↑n⌋₊, (n + i)}.Infinite

number theoryA3916686 sources

declared proved (Lean) · formalizederdos:464

True ↔ ∀ (n : ℕ → ℕ), StrictMono n → (∀ (k : ℕ), 0 < n k) → IsLacunary n → ∃ θ, Irrational θ ∧ ¬Dense (Set.range fun k => ↑(θ * ↑(n k)))

number theoryN/A4 sources

declared proved (Lean) · formalizederdos:469

Does the sum of the reciprocals of all primitive pseudoperfect numbers converge?

number theorydivisorsA006036 · A119425 · possible5 sources

declared proved (Lean) · formalizederdos:476

True ↔ ∀ (p : ℕ), Fact (Nat.Prime p) → ∀ (A : Finset (ZMod p)), A.restrictedSumset.card ≥ min (2 * A.card - 3) p

number theoryadditive combinatoricsN/A5 sources

declared proved (Lean) · formalizederdos:481

∀ {r : ℕ} (a b : Fin r → ℕ), (∀ (i : Fin r), 0 < a i) → 1 < ∑ i, 1 / ↑(a i) → ∃ k, ¬((Erdos481.T a b)^[k] [1]).Nodup

number theoryN/A5 sources

declared proved (Lean) · formalizederdos:484

∃ c, 0 < c ∧ ∀ (k : ℕ), 0 < k → ∃ N₀, ∀ (N : ℕ), N₀ ≤ N → ∀ (f : ℕ → Fin k), c * ↑N ≤ ↑{n ∈ Finset.Icc 1 N | ∃ a ∈ Finset.Icc 1 N, ∃ b ∈ Finset.Icc 1 N, a ≠ b ∧ f a = f b ∧ a + b = n}.card

number theoryadditive combinatoricsramsey theoryN/A5 sources

declared proved (Lean) · formalizederdos:487

True ↔ ∀ (A : Set ℕ), A.HasPosDensity → ∃ a ∈ A, ∃ b ∈ A, ∃ c ∈ A, a ≠ b ∧ b ≠ c ∧ a ≠ c ∧ a.lcm b = c

number theoryN/A5 sources

declared proved (Lean) · formalizederdos:493

True ↔ ∃ k N, ∀ (n : ℤ), N ≤ n → ∃ a, (∀ (i : Fin k), 2 ≤ a i) ∧ ∏ i, a i - ∑ i, a i = n

number theoryN/A5 sources

declared proved (Lean) · formalizederdos:498

True ↔ ∀ (n : ℕ) (z : Fin n → ℂ), (∀ (i : Fin n), 1 ≤ ‖z i‖) → ∀ (c : ℂ), {ε | (∀ (i : Fin n), ε i = -1 ∨ ε i = 1) ∧ ∑ i, ↑(ε i) * z i ∈ Metric.ball c 1}.ncard ≤ n.choose (n / 2)

combinatoricsanalysisN/A5 sources

declared proved (Lean) · formalizederdos:499

True ↔ ∀ (n : ℕ), ∀ M ∈ doublyStochastic ℝ (Fin n), ∃ σ, ↑n ^ (-↑n) ≤ ∏ i, M i (σ i)

combinatoricsN/A5 sources

declared proved (Lean) · formalizederdos:512

True ↔ ∃ c > 0, ∀ (N : ℕ) (A : Finset ℤ), A.card = N → c * Real.log ↑N ≤ ∫ (θ : ℝ) in 0..1, ‖∑ n ∈ A, additiveChar (↑n * θ)‖

analysisN/A4 sources

declared proved (Lean) · formalizederdos:519

True ↔ ∃ c, 0 < c ∧ ∀ (n : ℕ) (hn : 0 < n) (z : Fin n → ℂ), z ⟨0, hn⟩ = 1 → ∃ k, c < ‖Erdos519.powerSum z (↑k + 1)‖

analysisN/A5 sources

declared proved (Lean) · formalizederdos:532

True ↔ ∀ (c : ℕ → Fin 2), ∃ A, A.Infinite ∧ ∃ color, ∀ (S : Finset ℕ), S.Nonempty → ↑S ⊆ A → c (∑ n ∈ S, n) = color

number theoryramsey theoryN/A4 sources

declared proved (Lean) · formalizederdos:540

True ↔ ∃ C, 0 < C ∧ ∀ (N : ℕ), 0 < N → ∀ (A : Finset (ZMod N)), C * √↑N ≤ ↑A.card → Erdos540.HasZeroSubsetSum A

number theoryA0344635 sources

declared proved (Lean) · formalizederdos:541

True ↔ ∀ (p : ℕ), Fact (Nat.Prime p) → ∀ (a : Fin p → ZMod p), (∃ r, ∀ (S : Finset (Fin p)), S ≠ ∅ → ∑ i ∈ S, a i = 0 → S.card = r) → (Set.range a).ncard ≤ 2

number theoryN/A5 sources

declared proved (Lean) · formalized · $100erdos:582

True ↔ ∃ V x G, G.CliqueFree 4 ∧ Erdos582.EdgeRamseyTriangle G

graph theoryramsey theoryN/A5 sources

declared proved (Lean) · formalizederdos:618

True ↔ ∀ (G : (n : ℕ) → SimpleGraph (Fin n)), (∀ (n : ℕ), (G n).CliqueFree 3) → ((fun n => ↑(G n).maxDegree) =o[Filter.atTop] fun n => ↑n ^ (1 / 2)) → (fun n => ↑(Erdos618.h2 (G n))) =o[Filter.atTop] fun n => ↑n ^ 2

graph theoryN/A5 sources

declared proved (Lean) · formalizederdos:621

True ↔ ∀ (n : ℕ) (G : SimpleGraph (Fin n)) (a t : ℕ), IsGreatest {k | ∃ A ⊆ G.edgeFinset, A.card = k ∧ ∀ (x y z : Fin n), G.Adj x y → G.Adj y z → G.Adj x z → ({s(x, y), s(y, z), s(x, z)} ∩ A).card ≤ 1} a → IsLeast {k | ∃ T ⊆ G.edgeFinset, T.card = k ∧ ∀ (x y z : Fin n), G.Adj x y → G.Adj y z → G.Adj x z → (T ∩ {s(x, y), s(y, z), s(x, z)}).Nonempty} t → 4 * (a + t) ≤ n ^ 2

graph theoryN/A5 sources

declared proved (Lean) · formalizederdos:639

True ↔ ∀ᶠ (n : ℕ) in Filter.atTop, ∀ (C : Sym2 (Fin n) → Fin 2), {e | ¬e.IsDiag ∧ ∀ (x y : Fin n), e = s(x, y) → ¬∃ z, z ≠ x ∧ z ≠ y ∧ C s(x, z) = C e ∧ C s(y, z) = C e}.ncard ≤ n ^ 2 / 4

graph theoryramsey theoryN/A5 sources

declared proved (Lean) · formalizederdos:645

∀ (c : ℕ → Bool), ∃ x d, 0 < x ∧ x < d ∧ ∃ C, c x = C ∧ c (x + d) = C ∧ c (x + 2 * d) = C

number theoryadditive combinatoricsramsey theoryN/A5 sources

declared proved (Lean) · formalizederdos:646

True ↔ ∀ (S : Finset ℕ), (∀ p ∈ S, Nat.Prime p) → {n | ∀ p ∈ S, Even (padicValNat p n.factorial)}.Infinite

number theoryfactorialsN/A5 sources

declared proved (Lean) · formalizederdos:659

True ↔ ∃ A, (∀ (n : ℕ), (A n).card = n ∧ ∀ S ⊆ A n, S.card = 4 → 3 ≤ EuclideanGeometry.distinctDistances S) ∧ (fun n => ↑(EuclideanGeometry.distinctDistances (A n))) =O[Filter.atTop] fun n => ↑n / √(Real.log ↑n)

geometrydistancespossible6 sources

declared proved (Lean) · formalizederdos:674

True ↔ Erdos674.solutionSet.Nonempty

number theoryN/A4 sources

declared proved (Lean) · formalizederdos:678

True ↔ ∀ᶠ (k : ℕ) in Filter.atTop, {(m, n) | n + k ≤ m ∧ Finset.lcmInterval m (k + 1) < Finset.lcmInterval n k}.Nonempty

number theorypossible5 sources

declared proved (Lean) · formalizederdos:698

True ↔ ∃ h, Filter.Tendsto h Filter.atTop Filter.atTop ∧ ∀ (n i j : ℕ), 2 ≤ i → i < j → j ≤ n / 2 → h n ≤ (n.choose i).gcd (n.choose j)

number theorybinomial coefficientspossible5 sources

declared proved (Lean) · formalizederdos:728

Whether there are infinitely many integers

a, b, n

with

a, b \ge \varepsilon n

such that

a!\cdot b!

divides

n!\cdot(a+b-n)!

while

a+b

exceeds

n

by more than

C\cdot\log n

number theoryfactorialsN/A6 sources

declared proved (Lean) · formalizederdos:729

True ↔ ∀ C > 0, ∃ K ≥ 3, {(a, b, n) | a > 0 ∧ b > 0 ∧ n > 0 ∧ ↑a + ↑b > ↑n + C * Real.log ↑n ∧ ∀ (p : ℕ), Nat.Prime p → p > K → padicValNat p (↑n.factorial / (↑a.factorial * ↑b.factorial)).den = 0}.Infinite

number theoryfactorialsN/A6 sources

declared proved (Lean) · formalizederdos:750

True ↔ ∀ (f : ℕ → NNReal), Filter.Tendsto f Filter.atTop Filter.atTop → ∃ V G, G.chromaticNumber = ⊤ ∧ ∀ (m : ℕ) (S : Set V), 0 < m → S.ncard = m → ∃ I ⊆ S, G.IsIndepSet I ∧ ↑m / 2 - f m ≤ ↑I.ncard

graph theorychromatic numberN/A5 sources

declared proved (Lean) · formalizederdos:756

True ↔ (fun n => ↑n) =O[Filter.atTop] fun n => ↑(Erdos756.maxRichDistances n)

geometrydistancespossible5 sources

declared proved (Lean) · formalizederdos:760

True ↔ ∃ c > 0, ∀ (V : Type u_1) [Finite V] (G : SimpleGraph V) (m : ℕ), G.chromaticNumber = ↑m → ∃ H k, ↑k ≤ H.coe.cochromaticNumber ∧ c * ↑m / Real.log ↑m ≤ ↑k

graph theorychromatic numberN/A5 sources

declared proved (Lean) · formalizederdos:785

True ↔ ∀ (A B : Set ℕ), A.Infinite → B.Infinite → Erdos785.IsExactAdditiveComplement A B → Filter.Tendsto (fun x => ↑(Erdos785.counting A x) * ↑(Erdos785.counting B x) - ↑x) Filter.atTop Filter.atTop

additive combinatoricsN/A5 sources

declared proved (Lean) · formalizederdos:798

True ↔ (fun n => ↑(Erdos798.t n)) =o[Filter.atTop] fun n => ↑n

geometryA1164465 sources

declared proved (Lean) · formalizederdos:818

True ↔ ∀ (K : ℝ), 0 < K → ∃ C, 0 < C ∧ ∃ c, 0 < c ∧ ∀ (A : Finset ℤ), 2 ≤ A.card → ↑(A + A).card ≤ K * ↑A.card → c * ↑A.card ^ 2 / Real.log ↑A.card ^ C ≤ ↑(A * A).card

additive combinatoricsN/A5 sources

declared proved (Lean) · formalizederdos:844

True ↔ ∀ (N : ℕ), IsGreatest {k | ∃ A ⊆ Finset.Icc 1 N, (∀ a ∈ A, ∀ b ∈ A, ¬Squarefree (a * b)) ∧ A.card = k} (Erdos844.evenOrOddNonSquarefree N).card

number theoryintersecting familyN/A5 sources

declared proved (Lean) · formalizederdos:865

∃ C > 0, ∀ᶠ (N : ℕ) in Filter.atTop, ∀ A ⊆ Finset.Icc 1 N, ↑A.card ≥ 5 / 8 * ↑N + C → ∃ a ∈ A, ∃ b ∈ A, ∃ c ∈ A, a ≠ b ∧ a ≠ c ∧ b ≠ c ∧ a + b ∈ A ∧ a + c ∈ A ∧ b + c ∈ A

number theoryadditive combinatoricspossible5 sources

declared proved (Lean) · formalizederdos:898

∀ (A B C P L M N : EuclideanSpace ℝ (Fin 2)), AffineIndependent ℝ ![A, B, C] → P ∈ interior ((convexHull ℝ) {A, B, C}) → N ∈ affineSpan ℝ {A, B} → (affineSpan ℝ {P, N}).direction ⟂ (affineSpan ℝ {A, B}).direction → M ∈ affineSpan ℝ {B, C} → (affineSpan ℝ {P, M}).direction ⟂ (affineSpan ℝ {B, C}).direction → L ∈ affineSpan ℝ {C, A} → (affineSpan ℝ {P, L}).direction ⟂ (affineSpan ℝ {C, A}).direction → dist P A + dist P B + dist P C ≥ 2 * (dist P M + dist P N + dist P L)

geometryN/A5 sources

declared proved (Lean) · formalizederdos:904

True ↔ ∀ (V : Type u_1) [inst : Fintype V] (G : SimpleGraph V) [inst_1 : DecidableRel G.Adj], ∀ r ∈ Set.Icc 1 (Erdos904.n V), Erdos904.turanNumber (Erdos904.n V) r ≤ G.edgeFinset.card → ∃ s, G.IsNClique r s ∧ 2 * r * G.edgeFinset.card ≤ Erdos904.n V * ∑ v ∈ s, G.degree v

graph theoryN/A5 sources

declared proved (Lean) · formalizederdos:905

∀ (n : ℕ) (G : SimpleGraph (Fin n)) [inst : DecidableRel G.Adj], ↑n ^ 2 / 4 < ↑G.edgeFinset.card → ∃ e ∈ G.edgeFinset, ↑n / 6 ≤ ↑(G.trianglesContaining e).card

graph theoryN/A5 sources

declared proved (Lean) · formalizederdos:907

True ↔ ∀ (f : ℝ → ℝ), (∀ (h : ℝ), 0 < h → Continuous fun x => f (x + h) - f x) → ∃ g a, Continuous g ∧ (∀ (x y : ℝ), a (x + y) = a x + a y) ∧ f = g + a

analysisN/A5 sources

declared proved (Lean) · formalizederdos:914

∀ {r m : ℕ}, 2 ≤ r → 1 ≤ m → ∀ {V : Type u_1} [inst : Fintype V] (G : SimpleGraph V) [inst_1 : DecidableRel G.Adj], Fintype.card V = r * m → m * (r - 1) ≤ G.minDegree → ∃ K, (∀ (i : Fin m), G.IsNClique r (K i)) ∧ Pairwise fun i j => Disjoint (K i) (K j)

graph theoryN/A5 sources

declared proved (Lean) · formalizederdos:923

True ↔ ∀ (V : Type u_1) (n : ℕ), ∃ k, ∀ (G : SimpleGraph V), ↑k ≤ G.chromaticNumber → ∃ H ≤ G, ↑n ≤ H.chromaticNumber ∧ H.CliqueFree 3

graph theorychromatic numberN/A5 sources

declared proved (Lean) · formalizederdos:966

Let

k,r\geq 2

. Does there exist a set

A\subseteq \mathbb{N}

that contains no non-trivial arithmetic progression of length

k+1

, yet in any

r

-colouring of

A

there must exist a monochromatic non-trivial arithmetic progression of length

k

? Answered in the affirmative.

number theoryadditive combinatoricsramsey theoryN/A6 sources

declared proved (Lean) · formalizederdos:974

∀ {n : ℕ} [inst : NeZero n] (z : Fin n → ℂ), z 0 = 1 → {k | ∀ j < n - 1, ∑ i, z i ^ (k + j) = 0}.Infinite → Erdos974.IsTuranConfiguration z

analysisN/A5 sources

declared proved (Lean) · formalizederdos:997

True ↔ ∀ (α : ℝ), ¬Erdos997.IsWellDistributed fun n => Int.fract (α * ↑(Nat.nth Nat.Prime n))

analysisdiscrepancyprimesN/A6 sources

declared proved (Lean) · formalizederdos:1000

True ↔ ∃ n, StrictMono n ∧ 0 < n 0 ∧ Filter.Tendsto (Erdos1000.phiAvg n) Filter.atTop (nhds 0)

number theorydiophantine approximationN/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 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 proved (Lean) · formalizederdos:1022

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

combinatoricshypergraphspossible6 sources

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