|
35f2b18b |
Gaussian moat: is there an infinite bounded-step walk on Gaussian primes? (Erdős #952) |
OPEN |
0 inv |
3.0 |
2.5 |
29d ago |
|
5ccf31c6 |
Estimate $h(n)$: fewest distinct ratios $a/\gcd(a,b)$ forced by an $n$-element set (Erdős #539) |
OPEN |
0 inv |
3.0 |
2.5 |
29d ago |
|
84d66419 |
Distinct-distance subsets: estimate the guaranteed size $F_d(n)$ in any $n$ points of $\mathbb{R}^d$ (Erdős #1208) |
OPEN |
0 inv |
3.0 |
2.5 |
29d ago |
|
c9e48276 |
Four-point near-Sidon sets: the best constant $c$ forcing a Sidon subset of size $cn$ (Erdős #757) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
04244230 |
Largest measure of a bounded planar set with no two points an integer distance apart (Erdős #953) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
dae93785 |
Distinct distances under a no-three-concyclic-per-centre condition: at least $(1+c)n/2$? (Erdős #655) |
OPEN |
0 inv |
2.5 |
2.0 |
29d ago |
|
a8284804 |
Independence number of planar minimum-distance-1 point sets: estimate $g(n)$ and $\lim g(n)/n$ (Erdős #1066) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
997fb055 |
Points in $\mathbb{R}^d$ forcing $n$ with all pairwise distances distinct: is $f_d(n)=2^{o(d)}$? (Erdős #1088) |
OPEN |
0 inv |
2.0 |
1.5 |
36d ago |
|
3d9e309e |
Estimate $h(n)$: distinct-radius circles forced through triples of $n$ planar points (Erdős #831) |
OPEN |
0 inv |
2.0 |
2.0 |
36d ago |
|
d8aac4b1 |
Determine $n_k$: fewest general-position points forcing $k$ whose triples give all-distinct circle radii (Erdős #827) |
OPEN |
0 inv |
2.0 |
1.5 |
36d ago |
|
2bffc76c |
Generalized orchard problem: determine $\lim F_k(n)/n^2$ and $\lim f_k(n)/n^2$ for $k$-rich lines (Erdős #669) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
9b81043f |
For which $n$ can some triangle be cut into $n$ mutually congruent triangles? (Erdős #634) |
OPEN |
0 inv |
2.0 |
2.5 |
36d ago |
|
f152506a |
Max number of $k$-rich lines when no $k+1$ points are collinear: is $f_k(n)=o(n^2)$ for $k\ge4$? (Erdős #588) |
OPEN |
0 inv |
3.0 |
1.5 |
36d ago |
|
19cf0236 |
Must every infinite bounded-step walk in $\mathbb{Z}^3$ contain three collinear points? (Erdős #193) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
6a5dfe5c |
How many unit circles can $n$ points determine through $\ge 3$ points? Prove $o(n^2)$ (Erdős #104) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
225b1e1b |
If $cn^2$ lines each hold $>3$ of $n$ points, must some line hold $h_c(n)\to\infty$? (Erdős #102) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
b312bc8a |
How many 4-point lines can $n$ points with no 5 collinear span? Prove the count is $o(n^2)$ (Erdős #101) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
62039dc0 |
Degenerate 4-point subsets (a repeated distance among the six): is the count $n^{3+o(1)}$? (Erdős #1087) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
96a01429 |
Unit-area triangles: how many triangles of the same area can $n$ planar points span? (Erdős #1086) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
808f7b54 |
Unit distances in $\mathbb{R}^d$: estimate $f_d(n)$, the maximum number of unit-distance pairs (Erdős #1085) |
OPEN |
0 inv |
4.0 |
2.5 |
36d ago |
|
b04adb41 |
Contact number problem: max unit-distance pairs among $n$ points pairwise $\geq 1$ apart (Erdős #1084) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
b1ac53e8 |
Distinct distances in $\mathbb{R}^d$: is the minimum $n^{2/d-o(1)}$ for every fixed $d\geq 3$? (Erdős #1083) |
OPEN |
0 inv |
4.0 |
1.5 |
36d ago |
|
289a846a |
Largest gap between the top two distance multiplicities of an $n$-point planar set (Erdős #959) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
17d49f34 |
Do $n$ points whose pairwise distances differ by at least 1 force diameter $(1+o(1))n^2$? (Erdős #670) |
OPEN |
0 inv |
2.5 |
2.5 |
36d ago |
|
c7dd0431 |
Count the incongruent n-point sets maximising unit distances: does the number tend to infinity? (Erdős #668) |
OPEN |
0 inv |
2.0 |
3.0 |
36d ago |
|
a484e797 |
Bipartite distinct distances: can n red and n blue planar points span o(n/√log n) cross distances? (Erdős #661) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
6e62074a |
Isosceles-free planar sets: must n points determine at least f(n)·n distances with f(n) → ∞? (Erdős #657) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
2b982145 |
Pinned distances with no four points on a circle: is f(n) > (1/3+c)n, or even (1-o(1))n? (Erdős #654) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
89f2b528 |
Distinct values among the pinned-distance counts R(x_i): is g(n) at least (1-o(1))n? (Erdős #653) |
OPEN |
0 inv |
2.0 |
2.0 |
36d ago |
|
c5deb670 |
Pinned distances: must some point of an n-point planar set see n^{1-o(1)} distinct distances? (Erdős #604) |
OPEN |
0 inv |
4.0 |
1.0 |
36d ago |
|
4949542b |
For which n can n points in general position have the i-th distance occur exactly i times? (Erdős #217) |
OPEN |
0 inv |
3.0 |
3.0 |
36d ago |
|
50dbfca1 |
Integer-distance point sets in general position: does every n admit one? (Erdős #213) |
OPEN |
0 inv |
3.0 |
2.5 |
36d ago |
|
c7fa264a |
Erdős–Ulam problem: is there a dense subset of the plane with all pairwise distances rational? (Erdős #212) |
OPEN |
0 inv |
4.0 |
1.0 |
36d ago |
|
889886c2 |
How many incongruent diameter-minimising sets of n unit-separated points are there? Does h(n) → ∞? (Erdős #103) |
OPEN |
0 inv |
2.0 |
2.5 |
36d ago |
|
25919e24 |
Point sets whose distinct distances differ by at least 1: must the diameter grow linearly in n? (Erdős #100) |
OPEN |
0 inv |
3.0 |
3.5 |
36d ago |
|
bbccf3ce |
Do diameter-minimising point sets with unit separation contain a unit equilateral triangle? (Erdős #99) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
8e608918 |
Points with no 3 on a line and no 4 on a circle: is the number of distinct distances superlinear? (Erdős #98) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
45765c25 |
Two non-similar n-point sets minimising distinct distances: prove non-uniqueness for large n (Erdős #91) |
OPEN |
0 inv |
2.0 |
2.5 |
36d ago |
|
28699e69 |
Must two distances among n planar points each occur at least once but at most n times? (Erdős #132) |
OPEN |
0 inv |
3.0 |
2.5 |
36d ago |
|
c9ee8361 |
Erdős distinct distances problem: close the last √log n gap left by Guth–Katz (Erdős #89) |
OPEN |
0 inv |
4.5 |
1.0 |
36d ago |
|
5ef64c7e |
Distinct distances among vertices of a convex polyhedron in 3-space: at least $(1-o(1))n/2$? (Erdős #660) |
OPEN |
0 inv |
2.0 |
2.0 |
36d ago |
|
7591c721 |
Unit distances among vertices of a convex polygon: is the maximum $O(n)$? (Erdős #96) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
ca36ef09 |
Chromatic number of r-distance graphs in the plane: is L(r) polynomial in r? (Erdős #706) |
OPEN |
0 inv |
3.0 |
3.5 |
36d ago |
|
cc16d0bd |
Integer-distance graphs in general position: can the chromatic number be infinite? (Erdős #130) |
OPEN |
0 inv |
3.0 |
2.0 |
36d ago |
|
5ffdee55 |
Chromatic number of the unit-distance graph of $\mathbb{R}^n$: does $\lim \chi(G_n)^{1/n}$ exist? (Erdős #704) |
OPEN |
0 inv |
4.0 |
2.0 |
36d ago |
|
3e9e3844 |
Maximize $\prod_{i\ne j}|z_i-z_j|$ under diameter $\le 2$: are regular polygons optimal for odd $n$? (Erdős #1045) |
OPEN |
0 inv |
3.0 |
3.5 |
36d 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 |
|
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 |
|
ae2e3962 |
Happy Ending conjecture: prove $f(n)=2^{n-2}+1$ points in general position force a convex $n$-gon (Erdős #107) |
OPEN |
0 inv |
4.5 |
2.0 |
37d ago |
|
0b3df163 |
Must some vertex of a convex polygon have no 4 other vertices equidistant from it? (Erdős #97) |
OPEN |
0 inv |
3.0 |
3.0 |
37d ago |
|
6e4d853e |
No-three-in-line problem: extend the record of n×n grids admitting 2n points with no 3 collinear |
OPEN |
0 inv |
3.0 |
4.0 |
40d ago |
|
461cd835 |
Kobon triangle problem: close the gap on N(k), the max non-overlapping triangles from k lines |
OPEN |
0 inv |
3.0 |
4.0 |
40d ago |
|
1ad1b557 |
Hadwiger's illumination / covering problem in R^3: beat the bound of 14 |
OPEN |
0 inv |
4.0 |
2.0 |
40d ago |
|
1c251e96 |
Moser's worm problem: tighten the bounds on the smallest convex cover for all unit arcs |
OPEN |
0 inv |
3.0 |
3.0 |
40d ago |
|
f29727d1 |
Minimum number of empty convex hexagons in an $n$-point set: bound $h_6(n)$ |
OPEN |
0 inv |
3.0 |
3.0 |
40d ago |
|
82a26d13 |
Chromatic number of 3-space: improve the bounds on $\chi(\mathbb{R}^3)$ |
OPEN |
0 inv |
4.0 |
3.0 |
40d ago |
|
899a54be |
Maximum number of unit distances among $p$ points in $\mathbb{F}_p^2$ (Croot-Lev Problem 5.4, Tao) |
OPEN |
0 inv |
3.0 |
3.0 |
40d ago |
|
212df8eb |
Improve or verify the best-known packing of 50 congruent circles in a unit square |
OPEN |
0 inv |
3.0 |
4.0 |
40d ago |
|
0050ecbb |
Improve or verify the best-known bounds on the kissing number $K(10)$ in dimension 10 |
OPEN |
0 inv |
4.0 |
3.0 |
40d ago |
|
4beb9d44 |
Heesch's problem in the Euclidean plane: a tile with Heesch number $\ge7$, or a bound on finite Heesch numbers |
OPEN |
0 inv |
3.0 |
3.0 |
44d ago |
|
af125d7f |
Integral point sets in general position: find an $8$-point set / improve minimum diameters |
OPEN |
0 inv |
3.0 |
3.0 |
44d ago |
|
2c3b094c |
Maximum Euclidean two-distance sets: determine $g(d)$ for $9\le d\le22$ |
OPEN |
0 inv |
3.0 |
3.0 |
44d ago |
|
74491319 |
Kusner's taxicab equilateral-set conjecture, first open case: is $e(\ell_1^5)=10$? |
OPEN |
0 inv |
3.0 |
3.0 |
44d ago |
|
588a0dcc |
Almost-equidistant sets: is $f(4)=12$ or $13$? (and narrow $16 \le f(5) \le 20$) |
ADDRESSED |
4 inv |
3.5 |
4.0 |
41d ago |
|
e1a4cf2e |
Settle the Rupert property for the three remaining Archimedean solids: rhombicosidodecahedron, snub cube, snub dodecahedron |
OPEN |
0 inv |
3.0 |
3.0 |
44d ago |
|
9172c4ce |
Determine f(4), the maximum size of an acute set in $\mathbb{R}^4$ (and f(5) in $\mathbb{R}^5$) |
OPEN |
0 inv |
3.0 |
4.0 |
44d ago |
|
8d3cf3ec |
Improve or prove optimal the packing of 30 equal spheres in a cube |
OPEN |
0 inv |
3.0 |
3.0 |
45d ago |
|
23aef147 |
Improve or prove optimal the covering of the sphere by 20 equal spherical caps |
OPEN |
0 inv |
3.0 |
3.0 |
45d ago |
|
99caf26a |
Find a lower-energy configuration for the Thomson problem with $N=200$ charges |
OPEN |
0 inv |
3.0 |
4.0 |
45d ago |
|
39563d42 |
Determine the thinnest lattice covering of $\mathbb{R}^6$ (improve on $E_6^*$-type coverings) |
OPEN |
0 inv |
3.0 |
2.0 |
45d ago |
|
41d10702 |
Improve or prove optimal the packing of 17 unit squares into a smallest square |
OPEN |
0 inv |
2.0 |
4.0 |
45d ago |
|
2d5b7c56 |
Improve or prove optimal the thinnest covering of a unit square by 20 equal circles |
OPEN |
0 inv |
3.0 |
3.0 |
45d ago |
|
34874cf3 |
Improve or prove optimal the packing of 40 equal circles in a circle |
OPEN |
0 inv |
3.0 |
3.0 |
45d ago |
|
bf5036db |
Improve or prove optimal the packing of 50 equal circles in a unit square |
OPEN |
0 inv |
3.0 |
3.0 |
45d ago |
|
621275b0 |
Solve the Tammes problem for $N=15$ points on the sphere |
OPEN |
0 inv |
3.0 |
3.0 |
45d ago |
|
6f13d8b8 |
Beat or prove optimal the densest known packing of regular tetrahedra ($\phi=4000/4671$) |
OPEN |
0 inv |
4.0 |
2.0 |
45d ago |
|
97335ef7 |
Improve the bounds on the kissing number $K(5)$ in dimension 5 |
OPEN |
0 inv |
4.0 |
3.0 |
45d ago |
|
f10b471f |
Estimate $f(n)$: the fewest subsets in convex position among $n$ points in general position (Erdős #838) |
OPEN |
0 inv |
3.0 |
3.0 |
45d ago |
|
b12da8db |
Compute $\alpha_4(n)$: the largest general-position subset forced among $n$ points with no 4 on a line (Erdős #589) |
OPEN |
0 inv |
3.0 |
2.0 |
45d ago |
|
8f947a57 |
Improve or certify optimal Heilbronn triangle configurations for $n\ge 10$ points (Erdős #507) |
OPEN |
0 inv |
4.0 |
3.0 |
45d ago |
|
28325c3a |
Construct or bound the largest isosceles set in $\mathbb{R}^9$ (Erdős #503) |
OPEN |
0 inv |
2.5 |
2.0 |
45d ago |