Vela Mathematics Program: problems

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95 problems · 2/2Number SourcesGraph view

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 solvederdos:858

Erdős problem 858

number theoryprimitive setsN/A4 sources

declared provederdos:863

Erdős problem 863

number theorysidon setsadditive combinatoricsN/A3 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 solvederdos:896

Erdős problem 896

number theorypossible3 sources

declared disproved (Lean)erdos:927

Erdős problem 927

graph theorypossible3 sources

declared openerdos:935

Erdős problem 935

number theoryA057521 · A389244 · possible2 sources

declared proved (Lean)erdos:947

Erdős problem 947

number theorycovering systemsN/A4 sources

declared solvederdos:948

Erdős problem 948

number theoryramsey theoryN/A3 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 disprovederdos:960

Erdős problem 960

geometrypossible3 sources

declared openerdos:963

Erdős problem 963

number theorypossible2 sources

declared proved (Lean)erdos:964

Erdős problem 964

number theorydivisorsN/A4 sources

declared openerdos:976

Erdős problem 976

number theoryN/A2 sources

declared openerdos:983

Erdős problem 983

number theorypossible2 sources

declared provederdos:986

Erdős problem 986

graph theoryramsey theoryA000791 · A0594423 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

declared openerdos:1070

Erdős problem 1070

geometrypossible2 sources

declared solvederdos:1089

Erdős problem 1089

geometrydistancespossible3 sources

declared solvederdos:1091

Erdős problem 1091

geometrychromatic numberN/A3 sources

declared openerdos:1131

Erdős problem 1131

analysispolynomialsN/A2 sources

declared openerdos:1132

Erdős problem 1132

analysispolynomialsN/A2 sources

declared disproved (Lean) · £10erdos:1134

Erdős problem 1134

number theoryA1856613 sources

declared openerdos:1143

Erdős problem 1143

number theoryprimesN/A2 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:1183

Erdős problem 1183

combinatoricsramsey theorypossible2 sources

declared solvederdos:1187

Erdős problem 1187

number theoryprimesadditive combinatoricsN/A3 sources

declared openerdos:1194

Erdős problem 1194

additive combinatoricsadditive basissidon setspossible2 sources

declared solvederdos:1195

Erdős problem 1195

number theorypossible3 sources

declared disproved (Lean)erdos:1197

Erdős problem 1197

analysisN/A5 sources

declared solvederdos:1202

Erdős problem 1202

number theoryprimespossible3 sources

declared openerdos:1208

Erdős problem 1208

geometrydistancesA193838 · A271490 · possible2 sources

declared provederdos:1217

Erdős problem 1217

number theorydivisorsprimitive setsN/A3 sources

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