Vela Mathematics Program: problems

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declared open · formalized · $500erdos:1

∃ C > 0, ∀ (N : ℕ) (A : Finset ℕ), Erdos1.IsSumDistinctSet A N → N ≠ 0 → C * 2 ^ A.card < ↑N

number theoryadditive combinatoricsA2766612 sources

declared open · formalized · $5000erdos:3

True ↔ ∀ (A : Set ℕ), (¬Summable fun a => 1 / ↑↑a) → ∃ᶠ (k : ℕ) in Filter.atTop, ∃ S ⊆ A, S.IsAPOfLength ↑k

number theoryadditive combinatoricsarithmetic progressionsA003002 · A003003 · A003004 · A0030052 sources

declared open · formalizederdos:14

True ↔ ∀ (A : Set ℕ), ∀ ε > 0, Erdos14.almostSquareRoot ε =O[Filter.atTop] Erdos14.nonUniqueSumCount A

number theorysidon setsadditive combinatoricsA143824 · possible2 sources

declared open · formalized · $1000erdos:30

True ↔ ∀ ε > 0, (fun N => ↑(Erdos30.h N) - √↑N) =O[Filter.atTop] fun N => ↑N ^ ε

number theorysidon setsadditive combinatoricsA143824 · A227590 · A0030222 sources

declared open · formalizederdos:36

Erdos36.M 5 = 3

number theoryadditive combinatoricsA393584 · possible3 sources

declared open · formalized · $500erdos:39

True ↔ ∃ A, A.Infinite ∧ IsSidon A ∧ ∀ ε > 0, (fun x => ↑x ^ (1 / 2 - ε)) =O[Filter.atTop] fun N => ↑(Set.Icc 1 N ∩ A).ncard

number theorysidon setsadditive combinatoricsN/A2 sources

declared open · formalized · $500erdos:41

∀ (A : Set ℕ), Erdos41.NtupleCondition A 3 → A.Infinite → Filter.liminf (fun N => ↑(A ∩ Set.Icc 1 N).ncard / ↑N ^ (1 / 3)) Filter.atTop = 0

number theorysidon setsadditive combinatoricsN/A2 sources

declared open · formalizederdos:44

True ↔ ∀ N ≥ 1, ∀ A ⊆ Finset.Icc 1 N, IsSidon ↑A → ∀ ε > 0, ∃ M > N, ∃ B ⊆ Finset.Icc (N + 1) M, IsSidon (↑A ∪ ↑B) ∧ (1 - ε) * √↑M ≤ ↑(A ∪ B).card

number theorysidon setsadditive combinatoricsN/A2 sources

declared open · formalized · $250erdos:52

True ↔ ∀ (ε : ℝ), 0 < ε → ε < 1 → ∃ C, 0 < C ∧ ∀ (A : Finset ℤ), max ↑(A + A).card ↑(A * A).card ≥ C * ↑A.card ^ (2 - ε)

number theoryadditive combinatoricsA2639963 sources

declared open · formalized · $500erdos:138

W(k)

is the least

N

such that every two-colouring of

\{1, \dots, N\}

contains a monochromatic

k

-term arithmetic progression, must

W(k+1) - W(k) \to \infty

additive combinatoricsA0053464 sources

declared open · formalizederdos:141

sorry ↔ ∀ k ≥ 3, ∃ s, Erdos141.Set.IsAPAndPrimeProgressionOfLength s k

additive combinatoricsprimesarithmetic progressionsA0065602 sources

declared open · formalized · $10000erdos:142

∀ (k : ℕ), (fun N => ↑(Erdos142.r k N)) =Θ[Filter.atTop] sorry

additive combinatoricsarithmetic progressionsA003002 · A003003 · A003004 · A0030053 sources

declared open · formalizederdos:155

True ↔ ∀ k ≥ 1, ∀ᶠ (N : ℕ) in Filter.atTop, Erdos155.F (N + k) ≤ Erdos155.F N + 1

additive combinatoricssidon setsA143824 · A227590 · A0030222 sources

declared open · formalizederdos:160

have lower_bound := sorry; (lower_bound =O[Filter.atTop] fun n => ↑(Erdos160.erdos_160.h n)) ∧ ∀ c > 0, ((fun n => Real.exp (c * Real.log ↑n ^ (1 / 12))) =O[Filter.atTop] fun n => ↑(Erdos160.erdos_160.h n)) → ∀ c > 0, (fun n => Real.exp (c * Real.log ↑n ^ (1 / 12))) =o[Filter.atTop] lower_bound

additive combinatoricsarithmetic progressionspossible2 sources

declared open · formalizederdos:168

Erdos168.F 0 = 0

additive combinatoricsA004059 · A057561 · A094708 · A3864392 sources

declared openerdos:169

Erdős problem 169

additive combinatoricsarithmetic progressionsA005346

declared open · formalizederdos:170

Filter.Tendsto (fun N => ↑(Erdos170.F N) / √↑N) Filter.atTop (nhds sorry)

additive combinatoricsA0466932 sources

declared open · formalizederdos:172

True ↔ ∀ (n : ℕ) (color : ℕ → Fin n) (m : ℕ), ∃ A, A.card ≥ m ∧ ∃ c, ∀ (S : Finset ↥A), S.Nonempty → color (∑ x ∈ S, ↑x) = c ∧ color (∏ x ∈ S, ↑x) = c

additive combinatoricsramsey theoryN/A2 sources

declared openerdos:176

Let

N(k, \ell)

be the least

N

such that every

f : [N] \to \{-1, 1\}

has a

k

-term arithmetic progression

P

with

|\sum_{n \in P} f(n)| \ge \ell

. In particular, is

N(k, 2) \le C^k

additive combinatoricsarithmetic progressionsdiscrepancypossible3 sources

declared openerdos:187

Erdős problem 187

additive combinatoricsramsey theoryN/A

declared openerdos:201

Erdős problem 201

additive combinatoricsarithmetic progressionsA003002 · A003003 · A003004 · A003005 · possible

declared open · formalized · $100erdos:241

True ↔ Asymptotics.IsEquivalent Filter.atTop (fun N => ↑(Erdos241.f N 3)) fun N => ↑N ^ (1 / 3)

additive combinatoricssidon setsA3877042 sources

declared openerdos:271

Erdős problem 271

additive combinatoricsarithmetic progressionsA005487

declared open · formalizederdos:272

Asymptotics.IsEquivalent Filter.atTop (fun N => ↑(Erdos272.maxArithInterCard N)) sorry

additive combinatoricsarithmetic progressionspossible2 sources

declared openerdos:335

Erdős problem 335

number theoryadditive combinatoricsN/A2 sources

declared open · formalizederdos:340

∀ ε > 0, (fun n => √↑n / ↑n ^ ε) =O[Filter.atTop] fun n => ↑(Set.range Finset.greedySidon ∩ Set.Icc 1 n).ncard

number theoryadditive combinatoricssidon setsA080200 · A0052822 sources

declared openerdos:483

Erdős problem 483

number theoryadditive combinatoricsramsey theoryA030126

declared open · formalizederdos:749

True ↔ ∀ ε > 0, ∃ A, 1 - ε ≤ (A + A).lowerDensity ∧ (Nat.cast ∘ AdditiveCombinatorics.sumRep A) =O[Filter.atTop] fun n => 1

additive combinatoricsN/A3 sources

declared openerdos:787

Erdős problem 787

additive combinatoricspossible

declared openerdos:788

Erdős problem 788

additive combinatoricspossible3 sources

declared open · formalizederdos:789

(fun n => ↑(Erdos789.subsetSumThreshold n)) =Θ[Filter.atTop] sorry

additive combinatoricspossible2 sources

declared openerdos:790

Erdős problem 790

additive combinatoricspossible

declared openerdos:791

Erdős problem 791

additive combinatoricsA066063

declared openerdos:792

Erdős problem 792

additive combinatoricspossible2 sources

declared open · formalizederdos:817

True ↔ (fun n => 3 ^ n) =O[Filter.atTop] fun n => ↑(Erdos817.g 3 n)

additive combinatoricspossible2 sources

declared openerdos:819

Erdős problem 819

additive combinatoricspossible2 sources

declared openerdos:840

Erdős problem 840

additive combinatoricssidon setsN/A

declared openerdos:864

Erdős problem 864

number theorysidon setsadditive combinatoricsA3891822 sources

declared openerdos:866

Estimate the least excess

g_k(N)

forcing

k

integers whose pairwise sums all lie in a dense subset of

\{1, \dots, 2N\}

; in particular, determine the positive variant

h_4(n)

number theoryadditive combinatoricspossible5 sources

declared openerdos:875

Erdős problem 875

additive combinatoricsN/A2 sources

declared openerdos:876

Erdős problem 876

additive combinatoricsN/A

declared open · formalizederdos:1097

False ↔ ∃ C > 0, ∀ (A : Finset ℤ), ↑(Erdos1097.CommonDifferencesThreeTermAP A).ncard ≤ C * ↑A.card ^ (3 / 2)

number theoryadditive combinatoricsN/A3 sources

declared open · formalizederdos:1145

True ↔ Erdos1145.Erdos1145Prop

additive combinatoricsadditive basisN/A2 sources

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 · $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

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