|
842215e4 |
Cover the lemniscate $\{|f(z)|\le 1\}$ of any monic polynomial by discs of total radius $\le 2$ (Erdős #509) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
d0f47f8d |
Erdős–Szekeres products: the true order of $\log f(n)$ for $\min\max_{|z|=1}|\prod_i(1-z^{a_i})|$ (Erdős #256) |
OPEN |
0 inv |
3.0 |
2.5 |
36d ago |
|
613d24b0 |
Is the maximum size of a $B_3$ set in $\{1,\ldots,N\}$ asymptotic to $N^{1/3}$? (Erdős #241) |
OPEN |
0 inv |
3.0 |
2.5 |
36d ago |
|
a1808a63 |
Sidon sets: does $F(N+k)\le F(N)+1$ hold for every fixed $k$ and all large $N$? (Erdős #155) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
2852c84d |
Thresholds $r_k(d_1,d_2)$: bounded-gap sequences whose $k$-fold sumsets avoid lacunary sets (Erdős #1112) |
OPEN |
0 inv |
2.0 |
1.5 |
36d ago |
|
7c8bbe58 |
How small can the gaps in an infinite sum-free sequence be — is $a_{n+1}-a_n<n$ possible? (Erdős #876) |
OPEN |
0 inv |
3.0 |
1.5 |
36d ago |
|
36b0e32f |
Largest family of subsets of $\{1,\ldots,N\}$ whose pairwise intersections are nonempty APs (Erdős #272) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
92dc82d2 |
Stanley sequences: explicit structure and growth of the greedy 3-AP-free sequences $A(n)$ (Erdős #271) |
OPEN |
0 inv |
3.0 |
3.5 |
36d ago |
|
263f9456 |
Riddell's $G_k(N)$: the largest $k$-AP-free subset forced in any $N$ integers, versus $R_k(N)$ (Erdős #201) |
OPEN |
0 inv |
3.0 |
3.5 |
36d ago |
|
f7defeb7 |
Reciprocal-sum capacity $f(k)$ of $k$-AP-free sets: estimate it; is $f(k)/\log W(k)\to\infty$? (Erdős #169) |
ACTIVE |
1 inv |
3.0 |
3.5 |
24d ago |
|
536c821a |
Estimate $h(N)$: fewest colours on $\{1,\ldots,N\}$ so every 4-term AP sees at least 3 colours (Erdős #160) |
ACTIVE |
1 inv |
3.0 |
3.0 |
23d ago |
|
79b2bcf8 |
Prove an asymptotic formula for $r_k(N)$, the largest $k$-AP-free subset of $\{1,\ldots,N\}$ (Erdős #142) |
OPEN |
0 inv |
4.5 |
2.5 |
36d ago |
|
0100a513 |
Admissible sequences with disjoint $r$-fold sum sets: how small can the gaps $a_{n+1}-a_n$ be? (Erdős #875) |
OPEN |
0 inv |
2.0 |
2.0 |
36d ago |
|
80cdc7ce |
How many sums in $[1,N]$ can a set of $\lfloor N^{1/2}\rfloor$ integers produce? Estimate $f(N)$ (Erdős #819) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
3056c0d1 |
Subset sums with no $k$-term arithmetic progression: is $g_3(n)\gg 3^n$? (Erdős #817) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
2d9663a7 |
Sum-free subsets: how much bigger than $n/3$ can one always find? Estimate $f(n)$ (Erdős #792) |
OPEN |
0 inv |
4.0 |
2.0 |
36d ago |
|
1ef6006d |
Minimal additive 2-basis for $\{0,\ldots,n\}$: pin the constant in $g(n)^2\sim cn$ (Erdős #791) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
ca2c9007 |
Strongly sum-free subsets of every $n$-set: is $l(n)<n^{1-c}$, or is $l(n)\ge n^{1-o(1)}$? (Erdős #790) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
ac35354b |
Largest subset where equal sums force equally many summands: estimate $h(n)$ (Erdős #789) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
0ccdbc41 |
Choi's sum-avoiding set function: is $f(n)\le n^{1/2+o(1)}$? (Erdős #788) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
1491b2d7 |
Estimate $g(k)$: the least $n>k+1$ with all prime factors of $\binom{n}{k}$ exceeding $k$ (Erdős #1095) |
OPEN |
0 inv |
3.0 |
2.5 |
36d ago |
|
e44efcda |
Least prime factor of $\binom{n}{k}$: at most $\max(n/k,k)$ with finitely many exceptions? (Erdős #1094) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
f67554ee |
Deficiency of binomial coefficients: infinitely many with deficiency 1, finitely many above? (Erdős #1093) |
OPEN |
0 inv |
2.5 |
3.5 |
36d ago |
|
3d5f247b |
Is every multiplicity t realized by some repeated binomial coefficient? (Singmaster-type, Erdős #849) |
OPEN |
0 inv |
4.0 |
2.0 |
36d ago |
|
108aaf95 |
The least integer not dividing $\binom{2n}{n}$: pin down its typical growth rate (Erdős #731) |
OPEN |
0 inv |
2.0 |
3.5 |
36d ago |
|
60a0dc1e |
Powers of 2 with only digits 0 and 1 in base 3: are there finitely many? (Erdős #406) |
OPEN |
0 inv |
2.5 |
2.0 |
36d ago |
|
a3040e41 |
For every k, find n with $(n-k)(n-k+1)\cdots n$ dividing $\binom{2n}{n}$ (Erdős #396) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
6d252347 |
Is the sum of 1/p over primes p ≤ n not dividing $\binom{2n}{n}$ bounded uniformly in n? (Erdős #377) |
OPEN |
0 inv |
3.0 |
2.5 |
36d ago |
|
2306a439 |
Infinitely many $n\neq m$ with $\binom{2n}{n}$, $\binom{2m}{m}$ having the same prime divisors? (Erdős #730) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
7c83b02e |
Growth of $M_n(t)=\max_{x\in[-1,1]}|\sum_{k\le n}(-1)^{\epsilon_k(t)}x^k|$ for random signs (Erdős #524) |
OPEN |
0 inv |
3.0 |
2.5 |
36d ago |
|
10c87f43 |
Is the longest arithmetic progression of primes in $\{1,\ldots,N\}$ of length $o(\log N)$? (Erdős #200) |
OPEN |
0 inv |
3.0 |
2.5 |
36d ago |
|
897d61c4 |
Partition $\mathbb{N}$ into two sets, each permutable to avoid monotone 3-term APs (Erdős #197) |
OPEN |
0 inv |
2.0 |
2.0 |
36d ago |
|
cbd4950c |
Must every permutation of $\mathbb{N}$ contain a monotone 4-term arithmetic progression? (Erdős #196) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
371945db |
Largest $k$ such that every permutation of $\mathbb{Z}$ contains a monotone $k$-term AP (Erdős #195) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
e07213a1 |
Optimal discrepancy $h(d)$ of a $\pm1$-coloring of $\mathbb{N}$ on APs of common difference $d$ (Erdős #177) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
eb5cde27 |
Sums of distinct powers from several bases: the Burr–Erdős–Graham–Li completeness conjecture (Erdős #124) |
OPEN |
0 inv |
3.0 |
3.5 |
36d ago |
|
58b79afb |
Distinct common differences of 3-APs in an n-element integer set: pin down the maximal order (Erdős #1097) |
OPEN |
0 inv |
4.5 |
1.5 |
36d ago |
|
a0663382 |
Maximum size of a subset of $\{1,\ldots,N\}$ with at most one repeated pairwise sum (Erdős #864) |
OPEN |
0 inv |
2.0 |
3.0 |
36d ago |
|
9aa1b48f |
Growth of the Schur numbers f(k): is the least N forcing a monochromatic a+b=c exponential in k? (Erdős #483) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
cd6883a8 |
How long a monochromatic AP with difference $d$ does every 2-colouring of the integers force? (Erdős #187) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
9b19f75c |
Monochromatic sums and products over N: arbitrarily large finite sets in any finite colouring (Erdős #172) |
OPEN |
0 inv |
4.0 |
2.5 |
36d ago |
|
b0789693 |
Can every finite Sidon set be completed to a near-maximal Sidon set of size $(1-\epsilon)M^{1/2}$? (Erdős #44) |
OPEN |
0 inv |
3.5 |
2.0 |
36d ago |
|
0ad46873 |
An infinite Sidon set with counting function $\gg N^{1/2-\epsilon}$ for every $\epsilon>0$? (Erdős #39) |
OPEN |
0 inv |
4.0 |
1.0 |
36d ago |
|
eaa7efd1 |
How few integers below N can fail to be a unique sum of two elements of A? (Erdős #14) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
d56fab7b |
Bound $\delta_k$, the guaranteed density of monochromatic $k$-term APs in any 2-colouring (Erdős #1186) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
dcc23e24 |
Estimate $g_k(N)$: the surplus forcing all pairwise sums of some $k$ integers into $A$ (Erdős #866) |
OPEN |
0 inv |
2.0 |
3.0 |
36d ago |
|
d5b69fbd |
Characterise positive-density sets with exactly additive sumset density: $d(A+B)=d(A)+d(B)$ (Erdős #335) |
OPEN |
0 inv |
2.5 |
1.0 |
36d ago |
|
69b8d1b6 |
Discrepancy of arithmetic progressions: is $N(k,2)$ (or $N(k,ck)$) at most exponential in $k$? (Erdős #176) |
ACTIVE |
1 inv |
3.0 |
3.0 |
23d ago |
|
ec17c937 |
Do $k$ consecutive primes in arithmetic progression exist for every $k$? (Erdős #141) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
dabbc5cd |
Erdős–Szemerédi sum–product problem: is $\max(|A+A|,|AA|)\gg |A|^{2-\epsilon}$ for integer sets? (Erdős #52) |
OPEN |
0 inv |
4.5 |
2.0 |
36d ago |