Vela Mathematics Program: problems

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138 problems · 3/3Number SourcesGraph view

declared openerdos:953

What is the largest possible measure of a subset of a radius-

R

disk in

\mathbb{R}^2

containing no pair of points at a positive integer distance? A Poisson-Bessel kernel argument gives

M(R) \ll R^{1/2}

; with Sárközy's lower construction,

M(R) = R^{1/2 + o(1)}

geometrydistancesN/A3 sources

declared open · formalizederdos:959

How large can the difference between the largest and second-largest distance multiplicities be among

n

planar points?

geometrydistancesN/A4 sources

declared disprovederdos:960

Erdős problem 960

geometrypossible3 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 provederdos:986

Erdős problem 986

graph theoryramsey theoryA000791 · A0594423 sources

declared proved · formalizederdos:987

True ↔ ∀ (x : ℕ → ℝ), (∀ (j : ℕ), x j ∈ Set.Ioo 0 1) → Filter.limsup (fun k => Erdos987.A x k) Filter.atTop = ⊤

analysisdiscrepancyN/A4 sources

declared disproved (Lean) · formalizederdos:990

False ↔ ∃ C, ∀ (f : Polynomial ℂ), f.coeff 0 ≠ 0 → ∀ (α β : ℝ), 0 ≤ α → α ≤ β → β ≤ 2 * Real.pi → |↑(Erdos990.rootArgCount f (Set.Icc α β)) - (β - α) / (2 * Real.pi) * ↑f.natDegree| ≤ C * √(↑f.support.card * Real.log (Erdos990.M f))

analysisN/A6 sources

declared open · formalizederdos:996

Let

n_1<n_2<\cdots

be a lacunary sequence of integers and

f\in L^2([0,1])

with

n

th Fourier partial sum

f_n

. Is there an absolute constant

C>0

such that if

\| f-f_n\|_2 \ll (\log\log\log n)^{-C}

then

\frac{1}{N}\sum_{k\leq N}f(\{\alpha n_k\})\to\int_0^1 f

for almost every

\alpha

? A preprint answers this negatively via a dyadic spike-block counterexample.

analysisN/A4 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:1014

∀ (k : ℕ), 3 ≤ k → Filter.Tendsto (fun l => ↑R(k, l + 1) / ↑R(k, l)) Filter.atTop (nhds 1)

graph theorypossible7 sources

declared solved (Lean) · formalizederdos:1026

For a sequence of

n

distinct reals, determine the largest constant

c

such that some monotonic subsequence always has sum exceeding

(c-o(1))\cdot(1/\sqrt{n})

times the total sum. Resolved as

c = 1

combinatoricsA391431 · A3914906 sources

declared openerdos:1032

Do arbitrarily large 4-chromatic edge-critical graphs exist with minimum degree bounded below by a positive constant times the number of vertices?

graph theorychromatic numberpossible3 sources

declared open · formalizederdos:1038

Among all nonconstant monic polynomials

f

whose roots lie in

[-1, 1]

, determine

\inf_f |\{x \in \mathbb{R} : |f(x)| < 1\}|

analysisN/A4 sources

declared openerdos:1039

For

f(z) = \prod_{i=1}^n (z - z_i)

with all

|z_i| \le 1

, let

\rho(f)

be the radius of the largest disc contained in

\{z : |f(z)| < 1\}

. Is

\rho(f) \gg 1/n

? The worst case is now known to be

\Theta(1/n)

, with the explicit bound

\rho(f) \ge (\log 2)/n

analysisN/A3 sources

declared openerdos:1040

For a closed infinite set

F \subseteq \mathbb{C}

, let

\mu(F)

be the infimum of

|\{z : |f(z)| < 1\}|

over monic polynomials with zeros in

F

. Is

\mu(F)

determined only by the transfinite diameter of

F

analysisN/A3 sources

declared proved (Lean) · formalizederdos:1051

True ↔ ∀ (a : ℕ → ℤ), StrictMono a → Erdos1051.GrowthCondition a → Irrational (Erdos1051.ErdosSeries a)

irrationalityN/A6 sources

declared open · formalizederdos:1061

For

S(x) = \#\{(a,b) : a + b \le x,\ \sigma(a) + \sigma(b) = \sigma(a+b)\}

, is

S(x) \sim cx

? The preprint claims

S(x)

grows faster than

x (\log x)^R

for every fixed

R

, ruling out the linear asymptotic.

number theoryA110177 · possible4 sources

declared solvederdos:1089

Erdős problem 1089

geometrydistancespossible3 sources

declared solvederdos:1091

Erdős problem 1091

geometrychromatic numberN/A3 sources

declared disproved · formalizederdos:1092

False ↔ (fun n => ↑n) =o[Filter.atTop] fun n => ↑(Erdos1092.f 2 n)

geometrychromatic numberN/A5 sources

declared open · formalizederdos:1101

Does there exist a good pairwise-coprime sequence

u_n

with

\sum 1/u_n < \infty

and polynomial growth? What if one only requires

u_n \le e^{o(n)}

number theoryN/A4 sources

declared open · formalizederdos:1133

Must every sufficiently large node set admit bounded labels that force any polynomial fitting almost all labels at degree below

(1+\varepsilon)n

to have arbitrarily large uniform norm? Claimed via Beurling density for Bernstein spaces.

analysispolynomialsN/A4 sources

declared disproved (Lean) · formalizederdos:1138

False ↔ ∀ C > 1, Asymptotics.IsEquivalent Erdos1138.snd_gt_half_fst (Erdos1138.primeCount_Ioc_mul_const C) fun x => match x with | (x, y) => C * ↑(Erdos1138.sup_primeGap x) / Real.log y

number theoryprimespossible6 sources

declared disproved (Lean) · formalizederdos:1141

False ↔ Infinite ↑{n | Erdos1141.Erdos1141Prop n}

number theoryprimesA2145836 sources

declared proved (Lean) · formalizederdos:1148

True ↔ ∀ᶠ (n : ℕ) in Filter.atTop, Erdos1148.Erdos1148Prop n

number theoryA390380 · A3931686 sources

declared openerdos:1151

Let

\mathcal{L}^nf

be the Lagrange interpolation polynomials of a continuous

f

on the Chebyshev nodes. Prove that, for any closed

A\subseteq [-1,1]

, there exists a continuous function

f

such that

A

is the set of limit points of

\mathcal{L}^nf(x)

analysispolynomialsN/A3 sources

declared provederdos:1153

Erdős problem 1153

analysispolynomialsN/A3 sources

declared openerdos:1177

For a finite forbidden triple system

G

, what exact uncountable chromatic cardinalities occur among

G

-free triple systems, and how do those spectra interact? The revised manuscript answers the three exact-cardinal questions and claims a complete spectrum dichotomy.

set theorychromatic numberhypergraphsN/A3 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 · formalizederdos:1188

Estimate the number

F(x)

of minimal distinct covering systems whose moduli all lie in

[1, x]

. The candidate proof gives

\log\log F(x)/\log x \to 1

, i.e.

F(x) = \exp(x^{1+o(1)})

number theorycovering systemspossible4 sources

declared openerdos:1189

For irreducible covering sets of size

k

, determine their count, the possible largest modulus, the maximal reciprocal sum, and whether divisor-set examples occur infinitely often.

number theorycovering systemspossible2 sources

declared solved (Lean) · formalizederdos:1190

∀ (ε : ℝ), 0 < ε → ∀ᶠ (m : ℕ) in Filter.atTop, scaleL m ^ (-1 - ε) < Erdos1190.eps m ∧ Erdos1190.eps m < scaleL m ^ (-1 + ε)

number theorycovering systemspossible6 sources

declared solvederdos:1195

Erdős problem 1195

number theorypossible3 sources

declared proved (Lean) · formalizederdos:1196

Bounds the weighted sum

\sum 1/(a \log a)

taken over primitive sets of integers (sets where no element divides another).

number theoryprimitive setspossible6 sources

declared disproved (Lean)erdos:1197

Erdős problem 1197

analysisN/A5 sources

declared open · formalizederdos:1201

Is it true that for every

\epsilon,\eta>0

there exists a

k

such that the density of

n

for which

P(n(n+1)\cdots(n+k))>n^{1-\epsilon}

is at least

1-\eta

, where

P(m)

is the greatest prime divisor of

m

? A short argument via the Matomäki-Radziwiłł theorem establishes the lower-density version.

number theoryprimespossible4 sources

declared solvederdos:1202

Erdős problem 1202

number theoryprimespossible3 sources

declared provederdos:1217

Erdős problem 1217

number theorydivisorsprimitive setsN/A3 sources

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