Vela Mathematics Program: problems

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95 problems · 1/2Number SourcesGraph view

declared proved · $500erdos:21

Erdős problem 21

combinatoricsintersecting familyA3915992 sources

declared openerdos:81

Erdős problem 81

graph theorypossible2 sources

declared falsifiableerdos:106

f(n)

is the maximum total side length of

n

interior-disjoint squares packed in the unit square, is

f(k^2 + 1) = k

? An exact rational configuration packs

17

squares with total side length greater than

4

, refuting the identity at

k = 4

geometryN/A3 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 solved (Lean)erdos:192

Erdős problem 192

arithmetic progressionscombinatoricsN/A3 sources

declared disproved (Lean)erdos:231

Erdős problem 231

combinatoricsN/A3 sources

declared proved (Lean)erdos:237

Erdős problem 237

number theoryprimesN/A4 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 provederdos:380

Erdős problem 380

number theoryA070003 · A388654 · A387054 · A3891003 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 proved (Lean)erdos:433

Erdős problem 433

number theorypossible4 sources

declared disprovederdos:441

Erdős problem 441

number theoryA0685092 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 proved (Lean)erdos:490

Erdős problem 490

number theoryA3972054 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 disprovederdos:543

Erdős problem 543

number theorygroup theorypossible3 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 solvederdos:603

Erdős problem 603

combinatoricsset theoryN/A3 sources

declared proved (Lean)erdos:610

Erdős problem 610

graph theorypossible4 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 proved (Lean)erdos:658

Erdős problem 658

additive combinatoricspossible4 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 solvederdos:690

Erdős problem 690

number theorypossible4 sources

declared solved (Lean)erdos:696

Erdős problem 696

number theorydivisorspossible4 sources

declared proved (Lean)erdos:716

Erdős problem 716

graph theoryhypergraphspossible3 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 solved (Lean)erdos:765

Erdős problem 765

graph theoryturan numberA0068553 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 solvederdos:783

Erdős problem 783

number theoryN/A3 sources

declared openerdos:788

Erdős problem 788

additive combinatoricspossible3 sources

Problems · Vela Mathematics Program

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