Vela Mathematics Program: problems

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285 problems · 4/6Number 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 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 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 openerdos:773

Erdős problem 773

number theorysidon setssquaresA390813

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 openerdos:787

Erdős problem 787

additive combinatoricspossible

declared openerdos:788

Erdős problem 788

additive combinatoricspossible3 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 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 openerdos:813

Erdős problem 813

graph theorypossible

declared openerdos:819

Erdős problem 819

additive combinatoricspossible2 sources

declared openerdos:820

Erdős problem 820

number theoryA263647

declared openerdos:824

Erdős problem 824

number theorypossible

declared openerdos:827

Erdős problem 827

geometrypossible

declared openerdos:831

Erdős problem 831

geometrypossible

declared openerdos:836

Erdős problem 836

graph theoryhypergraphschromatic numberN/A2 sources

declared openerdos:837

Erdős problem 837

graph theoryhypergraphsN/A

declared openerdos:838

Erdős problem 838

geometryconvexpossible

declared openerdos:839

Erdős problem 839

number theoryN/A

declared openerdos:840

Erdős problem 840

additive combinatoricssidon setsN/A

declared openerdos:852

Erdős problem 852

number theoryprimesA001223 · A053597 · A0785152 sources

declared openerdos:854

Erdős problem 854

number theoryA389839 · A048670

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:860

Erdős problem 860

number theoryprimesA048670 · A058989

declared openerdos:864

Erdős problem 864

number theorysidon setsadditive combinatoricsA3891822 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

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