Vela Mathematics Program: problems

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59 problems · 1/2Number SourcesGraph view

declared openerdos:81

Erdős problem 81

graph theorypossible2 sources

declared open · $250erdos:165

Erdős problem 165

graph theoryramsey theoryA0007912 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:327

Erdős problem 327

number theoryunit fractionsA3849272 sources

declared openerdos:345

Erdős problem 345

number theorycomplete sequencesA0016612 sources

declared openerdos:367

Erdős problem 367

number theoryA0575213 sources

declared openerdos:368

Erdős problem 368

number theoryA0743993 sources

declared openerdos:374

Erdős problem 374

number theoryA388851 · A387184 · A389117 · A3891482 sources

declared openerdos:388

Erdős problem 388

number theoryN/A2 sources

declared openerdos:393

Erdős problem 393

number theoryfactorialsA3883022 sources

declared openerdos:411

Erdős problem 411

number theoryiterated functionsA383044 · possible2 sources

declared openerdos:415

Erdős problem 415

number theorypossible2 sources

declared openerdos:423

Erdős problem 423

number theoryA0052432 sources

declared openerdos:425

Erdős problem 425

number theorysidon setspossible2 sources

declared openerdos:432

Erdős problem 432

number theoryN/A2 sources

declared openerdos:451

Let

n_k

be the least integer greater than

2k

for which

\prod_{i=1}^k (n_k - i)

has no prime factor in

(k, 2k)

. How rapidly must

n_k

grow?

number theoryA3866203 sources

declared openerdos:460

Erdős problem 460

number theoryN/A2 sources

declared openerdos:514

For a transcendental entire function, how fast can

|f(z)|

be forced to grow along a path to infinity, and how short can such a path be in terms of the maximum modulus

M(r, f)

analysisN/A4 sources

declared openerdos:524

Erdős problem 524

analysisprobabilitypolynomialsN/A3 sources

declared openerdos:550

Let

m_1\leq\cdots\leq m_k

and

n

be sufficiently large. If

T

is a tree on

n

vertices and

G

is the complete multipartite graph with vertex class sizes

m_1,\ldots,m_k

, prove that

R(T,G)\leq (\chi(G)-1)(R(T,K_{m_1,m_2})-1)+m_1

graph theoryramsey theoryN/A4 sources

declared openerdos:616

Erdős problem 616

graph theoryN/A2 sources

declared openerdos:629

Erdős problem 629

graph theorychromatic numberpossible2 sources

declared openerdos:635

Erdős problem 635

number theoryN/A2 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 · $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:679

Erdős problem 679

number theoryN/A2 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: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: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 openerdos:776

Erdős problem 776

combinatoricspossible2 sources

declared openerdos:788

Erdős problem 788

additive combinatoricspossible3 sources

declared openerdos:819

Erdős problem 819

additive combinatoricspossible2 sources

declared openerdos:836

Erdős problem 836

graph theoryhypergraphschromatic numberN/A2 sources

declared openerdos:852

Erdős problem 852

number theoryprimesA001223 · A053597 · A0785152 sources

declared openerdos:856

Let

k\geq 3

and

f_k(N)

be the maximum of

\sum_{n\in A}\frac{1}{n}

over all

A\subseteq\{1,\ldots,N\}

containing no

k

subsets with the same pairwise least common multiple. Estimate

f_k(N)

. The claimed answer:

f_k(N)=(\log N)^{\gamma_k+o(1)}

, where

\gamma_k

is a weighted generalization of the Tang-Zhang sunflower capacity.

number theorypossible3 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:870

Let

k\geq 3

and

A

be an additive basis of order

k

. Does there exist a constant

c=c(k)>0

such that if

r(n)\geq c\log n

for all large

n

(where

r(n)

counts representations of

n

as a sum of at most

k

elements of

A

) then

A

must contain a minimal basis of order

k

? The claimed answer is no, for every

k\geq 3

number theoryadditive basisN/A3 sources

declared openerdos:875

Erdős problem 875

additive combinatoricsN/A2 sources

declared openerdos:935

Erdős problem 935

number theoryA057521 · A389244 · possible2 sources

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 openerdos:956

Erdős problem 956

geometrydistancesconvexpossible2 sources

declared openerdos:963

Erdős problem 963

number theorypossible2 sources

declared openerdos:976

Erdős problem 976

number theoryN/A2 sources

declared openerdos:983

Erdős problem 983

number theorypossible2 sources

declared openerdos:995

Erdős problem 995

analysisdiscrepancyN/A2 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 openerdos:1033

Erdős problem 1033

graph theorypossible2 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

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