Vela Mathematics Program: problems

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604 problems · 7/13Number SourcesGraph view

declared openerdos:612

Erdős problem 612

graph theoryN/A

declared openerdos:614

Erdős problem 614

graph theorypossible

declared openerdos:616

Erdős problem 616

graph theoryN/A2 sources

declared openerdos:620

Erdős problem 620

graph theorypossible

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 · $1000erdos:625

Erdős problem 625

graph theorychromatic numberN/A

declared openerdos:626

Erdős problem 626

graph theorychromatic numbercyclespossible

declared openerdos:627

Erdős problem 627

graph theorychromatic numberpossible

declared openerdos:629

Erdős problem 629

graph theorychromatic numberpossible2 sources

declared open · $25erdos:634

Erdős problem 634

geometrypossible

declared openerdos:635

Erdős problem 635

number theoryN/A2 sources

declared openerdos:638

Erdős problem 638

graph theoryramsey theoryN/A

declared openerdos:640

Erdős problem 640

graph theorychromatic numberN/A

declared openerdos:642

Erdős problem 642

graph theorycyclespossible

declared openerdos:643

Erdős problem 643

graph theoryhypergraphspossible

declared openerdos:644

Erdős problem 644

combinatoricspossible

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 openerdos:654

n

planar points have no four concyclic, must some point determine

(1 - o(1))n

distinct distances? Failing that, can one always force more than

(1/3 + c)n

geometrydistancespossible3 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 openerdos:657

Erdős problem 657

geometrydistancespossible

declared openerdos:660

Erdős problem 660

geometrydistancesconvexpossible

declared open · $50erdos:661

Erdős problem 661

geometrydistancespossible

declared openerdos:662

Among sufficiently large one-separated planar point sets, does the triangular lattice maximize the number of distances below each threshold? Explicit rational oblique lattices beat the triangular lattice under several closed- and strict-shell readings.

geometrydistancesN/A3 sources

declared openerdos:663

Erdős problem 663

number theoryA391668

declared openerdos:665

Erdős problem 665

combinatoricsN/A2 sources

declared openerdos:667

Erdős problem 667

graph theoryramsey theoryN/A

declared openerdos:668

Erdős problem 668

geometrydistancesA385657

declared openerdos:669

Erdős problem 669

geometryA003035 · A006065 · A008997 · possible

declared openerdos:670

Erdős asked whether every

n

-point set in Euclidean space whose pairwise distances are mutually at least 1 apart must have diameter at least

(1+o(1))n^2

. Disproved: an explicit high-dimensional construction beats the conjectured constant.

geometrydistancesN/A2 sources

declared open · $250erdos:671

For triangular arrays of nodes

a_i^n\in[-1,1]

let

\mathcal{L}^nf

be the Lagrange interpolation polynomials of a continuous

f

, with fundamental polynomials

p_i^n

. Is there a choice of nodes such that for every continuous

f

there is some

x

where

\limsup_n \sum_i\lvert p_{i}^n(x)\rvert=\infty

and yet

\mathcal{L}^nf(x) \to f(x)

? Is there a choice with

\limsup_n \sum_i\lvert p_{i}^n(x)\rvert=\infty

for every

x

, yet for every continuous

f

some

x

has

\mathcal{L}^nf(x)\to f(x)

? Both questions are claimed resolved in the affirmative.

analysisN/A3 sources

declared openerdos:675

Erdős problem 675

number theoryN/A2 sources

declared openerdos:676

Erdős problem 676

number theoryA390181 · in progress

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 openerdos:679

Erdős problem 679

number theoryN/A2 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 openerdos:684

For the least

k

at which the small-prime part of

\binom{n}{k}

exceeds

n^2

, how large can

f(n)

be?

number theoryprimesbinomial coefficientsA392019 · possible3 sources

declared openerdos:685

Erdős problem 685

number theoryprimesbinomial coefficientsN/A

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 · $1000erdos:687

Erdős problem 687

number theoryA048670 · A058989

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 openerdos:691

Erdős problem 691

number theoryN/A

declared openerdos:693

Erdős problem 693

number theorydivisorsA391118 · possible

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

declared openerdos:704

Erdős problem 704

graph theorygeometrychromatic numberN/A

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