|
ec8fdb76 |
The minimum overlap problem: pin down Erdős's constant $c$, now trapped in $(0.379005, 0.380876)$ (Erdős #36) |
OPEN |
0 inv |
3.0 |
4.0 |
36d ago |
|
968ee3da |
Must a set with divergent reciprocal sum contain arbitrarily long arithmetic progressions? (Erdős #3) |
OPEN |
0 inv |
4.5 |
1.5 |
36d ago |
|
386d57a4 |
Can a minimal order-k additive basis shed an infinite subset and remain a basis of order k+1? (Erdős #881) |
OPEN |
0 inv |
2.0 |
1.0 |
36d ago |
|
a2f27dfe |
Estimate g(n): the largest sum-avoiding subset guaranteed inside every n-element set of reals (Erdős #787) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
daeb07d1 |
A set with bounded representation function whose sumset has lower density 1−ε: does it exist? (Erdős #749) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
5215b46d |
Sparse rulers: determine the limit of F(N)/√N for minimal difference bases of {0,...,N} (Erdős #170) |
OPEN |
0 inv |
3.0 |
2.5 |
36d ago |
|
b3eeaef4 |
The maximal density of sets avoiding {n,2n,3n}: evaluate the limit and decide irrationality (Erdős #168) |
OPEN |
0 inv |
3.0 |
3.5 |
36d ago |
|
ad0ed6ee |
Growth of van der Waerden numbers: prove or disprove W(k)^{1/k} → ∞ (Erdős #138) |
OPEN |
0 inv |
4.5 |
2.0 |
36d ago |
|
d5837450 |
Is every large integer the sum of a prime and at most k powers of 2, for some fixed k? (Erdős #10) |
OPEN |
0 inv |
3.0 |
3.5 |
36d ago |
|
ec7b6900 |
Restricted order of an additive basis: existence, boundedness in the order, and equality (Erdős #338) |
OPEN |
0 inv |
3.0 |
2.0 |
37d ago |
|
76b88fe7 |
Exact order versus order of additive bases: evaluate $\lim_r h(r)/r^2$, and determine $h(4)$ (Erdős #336) |
OPEN |
0 inv |
3.0 |
2.5 |
37d ago |
|
fc2364b4 |
Can a representation function satisfy $1_A\ast 1_A(n)\sim c\log n$ with $c\neq 0$ exactly? (Erdős #66) |
OPEN |
0 inv |
3.5 |
1.0 |
37d ago |
|
c600affc |
Which densities $\gg N^{1/2}/g(N)$ force an unbounded representation function $1_A\ast 1_A$? (Erdős #40) |
OPEN |
0 inv |
3.5 |
1.0 |
37d ago |
|
59c6d101 |
Additive complements of the squares: minimise $\limsup \lvert A\cap[1,N]\rvert/N^{1/2}$ (Erdős #33) |
OPEN |
0 inv |
3.0 |
2.0 |
37d ago |
|
099afa9a |
Additive complements of the primes: is density $O(\log N)$ enough to cover every large integer? (Erdős #32) |
OPEN |
0 inv |
3.0 |
1.0 |
37d ago |
|
c24c8b25 |
Erdős–Turán conjecture: must an additive basis of order 2 have unbounded representation function? (Erdős #28) |
OPEN |
0 inv |
4.5 |
2.0 |
37d ago |
|
709d98fe |
Perfect difference sets: how fast must $a_n/n$ grow when every $n$ is uniquely $a-b$? (Erdős #1194) |
OPEN |
0 inv |
3.0 |
2.5 |
37d ago |
|
85ca6554 |
Does every order $r\geq 2$ admit an additive basis with $\sum_{n\leq x}f_r(n)^2\ll x$? (Erdős #1192) |
OPEN |
0 inv |
3.0 |
1.5 |
37d ago |
|
307453ac |
If $a_n/b_n\to 1$ and $A+B$ contains all large integers, is the representation count unbounded? (Erdős #1145) |
OPEN |
0 inv |
3.5 |
1.0 |
37d ago |
|
15a43cd1 |
Do $n/2$ vertices of degree $\geq n/2$ force every tree on $\leq n/2$ vertices? (Erdős #580) |
OPEN |
1 inv |
3.0 |
2.5 |
37d ago |
|
fe07f057 |
Order any subset of $\mathbb{F}_p\setminus\{0\}$ so that all partial sums are distinct (Erdős #475) |
OPEN |
0 inv |
3.5 |
3.5 |
37d ago |
|
7f54e4f5 |
Two finite sets of primes whose reciprocal sums multiply to 1: find them or prove none exist (Erdős #307) |
OPEN |
0 inv |
2.5 |
2.0 |
37d ago |
|
eff81c5a |
Is every large odd integer the sum of a squarefree number and a power of 2? (Erdős #11) |
OPEN |
0 inv |
3.0 |
2.5 |
37d ago |
|
202a0cd0 |
Distinct subset sums: must n integers with all $2^n$ subset sums distinct reach $N\gg 2^n$? (Erdős #1) |
OPEN |
0 inv |
4.5 |
2.0 |
37d ago |
|
822be9d3 |
Colour k-subsets of [2k] with k+1 colours so every (k+1)-set is rainbow: possible for k>2? (Erdős #835) |
OPEN |
0 inv |
2.0 |
3.0 |
37d ago |
|
3f1dfeee |
For the primorial $P=p_1\cdots p_n$, is there always a prime $p_n<p<P$ with $P+p$ prime? (Erdős #779) |
OPEN |
0 inv |
2.0 |
3.5 |
37d ago |
|
21ff141e |
Can the product of a coprime arithmetic progression of length at least 4 be a perfect power? (Erdős #672) |
OPEN |
0 inv |
3.5 |
1.5 |
37d ago |
|
19e31ed0 |
Is there an $n>24$ with $m+\tau(m)\leq n+2$ for every $m<n$? (Erdős #647) |
OPEN |
1 inv |
3.0 |
2.5 |
37d ago |
|
22745fee |
Do three consecutive powerful numbers exist? (Erdős #364) |
OPEN |
0 inv |
3.0 |
3.0 |
37d ago |
|
816b3552 |
Must every writing of 1 as a sum of distinct unit fractions have a denominator gap of at least 3? (Erdős #287) |
OPEN |
0 inv |
3.0 |
3.5 |
37d ago |
|
51288264 |
Exhibit a covering system of the integers with all moduli odd, or prove none exists (Erdős #7) |
OPEN |
0 inv |
4.5 |
2.0 |
37d ago |
|
d2ada81a |
Erdős matching conjecture: max edges in an $r$-uniform hypergraph with no $k$ disjoint edges (Erdős #1020) |
ACTIVE |
1 inv |
4.0 |
3.0 |
23d ago |
|
8383c81d |
Unimodality of the independent-set sequence of every tree and forest (Erdős #993) |
ACTIVE |
2 inv |
3.0 |
3.0 |
24d ago |
|
4694be38 |
Tree packing conjecture: do trees $T_2,\ldots,T_n$ with $|T_k|=k$ decompose $K_n$? (Erdős #743) |
ACTIVE |
1 inv |
4.0 |
3.0 |
24d ago |
|
c612c9e6 |
Balanced $r$-colourings of $K_{r^2+1}$: must some $K_{r+1}$ miss a colour? (Erdős #617) |
OPEN |
0 inv |
3.0 |
3.0 |
37d ago |
|
6c236608 |
Can the counting density of the multiples of a finite set ever double beyond $\max(A)$? (Erdős #488) |
OPEN |
0 inv |
2.5 |
3.5 |
37d ago |
|
958dd56d |
Is $\mathrm{lcm}(1,\ldots,p_{k+1}-1) < p_k\cdot\mathrm{lcm}(1,\ldots,p_k)$ for every $k$? (Erdős #458) |
OPEN |
0 inv |
3.0 |
3.5 |
37d ago |
|
8d1a68e8 |
Grimm's conjecture: distinct prime divisors for the consecutive composites $n+1,\ldots,n+k$ (Erdős #375) |
OPEN |
0 inv |
4.0 |
3.5 |
37d ago |
|
01e64dd0 |
Szemerédi's conjecture: n points with no 3 collinear determine at least n/2 distinct distances (Erdős #1082) |
OPEN |
0 inv |
3.5 |
2.5 |
37d ago |
|
6230b286 |
Erdős–Sós conjecture: (k-1)n/2 + 1 edges force every tree on k+1 vertices (Erdős #548) |
OPEN |
0 inv |
4.0 |
2.0 |
37d ago |
|
1d980793 |
Brocard–Ramanujan: are n = 4, 5, 7 the only solutions of n! = x^2 - 1? (Erdős #398) |
OPEN |
0 inv |
4.0 |
2.5 |
37d ago |
|
2e762fb0 |
Tuza's conjecture: delete 2k edges to kill all triangles when only k are edge-disjoint (Erdős #167) |
OPEN |
0 inv |
3.0 |
3.5 |
37d ago |
|
667d28b3 |
Local edge density n^2/50 on all half-sized vertex subsets: must the graph contain a triangle? (Erdős #128) |
OPEN |
0 inv |
3.0 |
2.0 |
37d ago |
|
335b7ef1 |
Packing k^2+1 squares in a unit square: is the maximum total side-length exactly k? (Erdős #106) |
OPEN |
0 inv |
3.0 |
3.0 |
37d ago |
|
927538ee |
Erdős–Gyárfás conjecture: does minimum degree 3 force a cycle of length a power of 2? (Erdős #64) |
OPEN |
0 inv |
4.0 |
3.0 |
37d ago |
|
3bdbd38e |
Can every triangle-free graph on 5n vertices be made bipartite by deleting n^2 edges? (Erdős #23) |
OPEN |
0 inv |
3.0 |
2.5 |
37d ago |
|
d006fcbf |
Short paths in lemniscates: are two roots always joined by a path of length < 2 in $\{|f|<1\}$? (Erdős #1041) |
ACTIVE |
1 inv |
3.0 |
2.5 |
15d ago |
|
cb372728 |
Does some vertex of a convex $n$-gon see at least $\lfloor n/2\rfloor$ distinct distances? (Erdős #982) |
OPEN |
0 inv |
3.0 |
2.5 |
37d ago |
|
bfb79f2f |
Prime power conjecture: does a finite projective plane of order $n$ force $n$ to be a prime power? (Erdős #723) |
OPEN |
0 inv |
4.0 |
2.0 |
37d ago |
|
1332eefd |
Do $\binom{n}{i}$ and $\binom{n}{j}$ always share a prime factor $p \ge i$? (Erdős #699) |
ACTIVE |
1 inv |
3.0 |
3.5 |
28d ago |