Vela Mathematics Program: problems

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604 problems · 8/13Number SourcesGraph view

declared openerdos:706

Erdős problem 706

graph theorychromatic numberpossible

declared open · $100erdos:708

Erdős problem 708

number theorypossible

declared openerdos:709

How long must an interval be to contain distinct representatives

x_i

, with

a_i \mid x_i

, for every

n

-element set of moduli

A = \{a_1, \dots, a_n\}

number theorypossible2 sources

declared open · ₹2000erdos:710

Erdős problem 710

number theoryA390246

declared open · ₹1000erdos:711

Erdős problem 711

number theorypossible

declared open · $500erdos:712

Erdős problem 712

graph theoryturan numberhypergraphspossible

declared open · $500erdos:713

Erdős problem 713

graph theoryturan numberN/A

declared openerdos:714

Erdős problem 714

graph theoryturan numberpossible

declared openerdos:719

Erdős problem 719

graph theoryhypergraphspossible

declared openerdos:724

Erdős problem 724

combinatoricsA001438

declared openerdos:725

Erdős problem 725

combinatoricsA001009

declared open · formalizederdos:726

True ↔ Asymptotics.IsEquivalent Filter.atTop (fun n => ∑ p ∈ Finset.range (n + 1) with Nat.Prime p ∧ ↑p / 2 < ↑n % ↑p, 1 / ↑p) fun n => Real.log (Real.log ↑n) / 2

number theoryN/A3 sources

declared open · formalizederdos:727

True ↔ ∀ k ≥ 2, {n | (n + k).factorial ^ 2 ∣ (2 * n).factorial}.Infinite

number theoryfactorialsA002503 · A343507 · A3893962 sources

declared open · formalizederdos:730

True ↔ Erdos730.S.Infinite

number theorybinomial coefficientsbase representationsA1295154 sources

declared openerdos:731

Let

A(n)

be the least positive integer not dividing

\binom{2n}{n}

. Erdos asked for the behaviour of

A(n)

for reasonable

n

. Under an explicit dyadic-regularity formalization of reasonable, the distribution is determined on dyadic intervals against the scale

F_X = \sqrt{2}(\log 2)^{1/4} L^{1/4} \exp\sqrt{(\log 2)L}

with

L = \log(2X)

number theorybinomial coefficientsA0061974 sources

declared openerdos:734

Erdős problem 734

combinatoricspossible2 sources

declared openerdos:738

Erdős problem 738

graph theorychromatic numberN/A

declared open · formalizederdos:740

True ↔ ∀ (V : Type u_1) (G : SimpleGraph V), Cardinal.aleph0 ≤ G.chromaticCardinal → ∀ (r : ℕ), ∃ H, H.coe.chromaticCardinal = G.chromaticCardinal ∧ Erdos740.NoShortOddCycle H.coe r

graph theorychromatic numberN/A2 sources

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 open · formalizederdos:757

∀ {A : Set ℝ}, sorry = sSup {c | Erdos757.IsAdmissible c}

geometrydistancessidon setspossible2 sources

declared openerdos:761

Erdős problem 761

graph theorychromatic numberN/A

declared openerdos:766

Erdős problem 766

graph theoryturan numberpossible

declared openerdos:768

Let

A(x)

count

n \le x

such that every prime

p \mid n

has a divisor

d > 1

n

with

d \equiv 1 \pmod p

. Erdos asked whether

A(x)/x = \exp(-(c+o(1))\sqrt{\log x}\log\log x)

. It does, with

c = 1/(2\sqrt{\log 2})

number theoryA001034 · A3522873 sources

declared open · formalizederdos:769

For the least cutoff

c(n)

after which every

k

occurs as the number of homothetic cubes in a decomposition of the unit

n

-cube, is

c(n) \gg n^n

? The Lean proof shows

c(n) = o(n^n)

along odd dimensions.

number theorygeometryA014544 · possible4 sources

declared open · formalizederdos:770

∀ {n : ℕ}, 2 < n → (Erdos770.h n = ↑n + 1 ↔ Nat.Prime (n + 1))

number theoryA263647 · possible2 sources

declared openerdos:773

Erdős problem 773

number theorysidon setssquaresA390813

declared open · formalizederdos:774

True ↔ ∀ (A : Set ℕ), A.Infinite → A.IsProportionatelyDissociated → ∃ T, (∀ S ∈ T, S.IsDissociated) ∧ T.Finite ∧ A = ⋃₀ T

number theoryN/A2 sources

declared openerdos:776

Erdős problem 776

combinatoricspossible2 sources

declared openerdos:778

Erdős problem 778

graph theoryN/A

declared openerdos:782

Erdős problem 782

number theoryN/A

declared open · formalizederdos:786

Erdos786.consecutivePrimesFrom 2 1 = {2, 3}

number theoryA143301 · possible2 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:796

g_3(n)

is the largest size of

A \subseteq [1,n]

with fewer than three representations of every product

a_1 a_2

, does its conjectured second-order normalized term converge? The candidate proof gives an explicit limit constant.

number theorypossible5 sources

declared openerdos:802

Erdős problem 802

graph theoryN/A

declared openerdos:805

Erdős problem 805

graph theorypossible

declared openerdos:809

Erdős problem 809

graph theoryramsey theorypossible

declared openerdos:810

Erdős problem 810

graph theoryramsey theorypossible

declared openerdos:811

Erdős problem 811

graph theoryramsey theorypossible

declared open · formalizederdos:812

True ↔ ∃ c > 0, ∀ᶠ (n : ℕ) in Filter.atTop, ↑(Combinatorics.hypergraphRamsey 2 (n + 1)) / ↑(Combinatorics.hypergraphRamsey 2 n) ≥ 1 + c

graph theoryramsey theoryA0594422 sources

declared openerdos:813

Erdős problem 813

graph theorypossible

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:820

Erdős problem 820

number theoryA263647

declared open · formalizederdos:821

True ↔ ∀ ε > 0, {n | ↑(Erdos821.g n) > ↑n ^ (1 - ε)}.Infinite

number theoryA0141972 sources

declared openerdos:824

Erdős problem 824

number theorypossible

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