Are growth rates of hyperbolic Coxeter groups Perron numbers?
Conjecture (Kellerhals–Perren): the growth rate of every cocompact hyperbolic Coxeter group is a Perron number. Testable polytope by polytope with CoxIter.
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Open problems in geometry
Packing, covering and discrete geometry problems where better configurations can be checked exactly, and where formal statements pin down what a proof must show.
The kissing number in dimension 5
Determine τ5, the maximum number of non-overlapping unit spheres touching a central unit sphere in R^5. Currently 40 ≤ τ5 ≤ 44.
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Dol'nikov's conjecture on piercing translates in the plane
Three finite families of translates of a planar convex body, with every two sets from different families intersecting: can one of the families always be pierced by 3 points?
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Is the flip graph of triangulations connected in dimensions 3 and 4?
Any two triangulations of a planar point set are connected by flips (Lawson); in dimension 5 and up this fails (Santos). For point sets in R³ and R⁴ the question is open.
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Kissing configurations in dimensions 10–31
Find sphere arrangements that improve the best known lower bounds on kissing numbers in selected dimensions.
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Radon-type partitions for unions of convex sets
What is the least f(d, s, t) such that any f points in R^d split into A ∪ B with every union of s convex sets covering A meeting every union of t convex sets covering B? Known: O(dst log(st+1)), and f(2, s, s) ≥ s².
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A uniform dilation constant for unimodular triangulations
Is there c_d such that c_d·P has a unimodular triangulation for every lattice d-polytope P? Knudsen–Mumford–Waterman give a dilation depending on P.
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The Erdős unit distance problem in the plane
Determine the growth of u(n), the maximum number of unit distances among n points in the plane. Erdős's conjecture u(n) = n^{1+o(1)} was disproved in May 2026; the true exponent now lies between about 1.014 (Sawin) and 4/3 (Spencer–Szemerédi–Trotter).
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The Hodge conjecture
Prove that on a non-singular complex projective variety every rational Hodge class is a rational linear combination of classes of algebraic cycles (Clay Millennium Prize Problem). A full solution is not expected here.
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Ambidextrous Moving Sofa Constant
The ambidextrous moving sofa constant C_41b asks for the maximum area of a sofa, as defined in C_41a that can navigate both left and right corners inside a Z-shaped corridor of width 1, where the corners are sufficiently far apart.
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Block Stacking Problem
Let n ≥ 1. Let C(n) be the largest displacement that the n^th block in a stack of identical rigid rectangular blocks of width 1 can be displaced horizontally over the edge of a table, with the stack remaining stable.
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Equidistant points in convex polygons
Is it true that every convex polygon has a vertex with no other 4 vertices equidistant from it?
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Erdős squares in a square problem
For any natural n, let C(n) denote the maximum possible sum of side lengths of n squares with disjoint interiors contained inside a unit square. Obtain upper and lower bounds for C(n) that are as strong as possible.
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Favard-length decay exponent
Let E⊂ ℝ^2 be a planar set. The Favard length of E is defined by Fav(E) := 1/π∫_0^π lvert Proj R_θ Ervert dθ, where Proj is orthogonal projection to the horizontal axis and R_θ is rotation by angle θ.
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Flatness constant in dimension 3
A convex body K⊂ℝ^d is hollow (lattice-free) with respect to a lattice Λ if int(K)∩Λ=∅. <a href="#CS2019-hollow-def">[CS2019-hollow-def]</a> For a hollow body, the lattice width is w(K) := min_u∈ℤ^d∖0 (max_x∈ Ku· x-min_x∈ Ku· x).
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Gilbert-Pollak conjecture (Steiner ratio)
C_43 is defined as the infimum of the ratio of the length of the Steiner Minimal Tree to the length of the Euclidean Minimum Spanning Tree over all finite sets of points V ⊆ ℝ^2: C_43 = inf_VL_S(V)/L_M(V), where L_S(V) and L_M(V) denote the lengths of Steiner Minimal Tree and Minimum Spanning Tree…
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Hadwiger covering / illumination number in ℝ^3
C_39=H_3 is the Hadwiger covering number in dimension 3, which can also be formulated in terms of illumination of the boundary.
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Heilbronn problem in a fixed bounding box
For any n ≥ 3 and any convex body K in the plane, let C(n,K) be the largest quantity such that in every configuration of n points in K, there exists a triple of points determining a triangle of area at most C(n,K) times the area of K. Establish upper and lower bounds on C(n,K).
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Heilbronn problem in an arbitrary convex bounding box
For any n ≥ 3 let C(n) be the largest quantity such that in every configuration of n points in the plane, there exists a triple of points determining a triangle of area at most C(n) times the area of their convex hull. Establish upper and lower bounds on C(n).
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Kakeya needle problem
Let n ≥ 2. Let C^T(n) denote the minimal area |bigcup_j=1^n T_j| of a union of triangles T_j with vertices (x_j,0), (x_j + 1/n, 0), (x_j + j/n, 1) for some real numbers x_1,…,x_n, and similarly define C^P(n) denote the minimal area |bigcup_j=1^n P_j| of a union of parallelograms P_j with vertices…
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Lebesgue universal covering constant
C_13b = a is the infimal area of a convex planar set Ω that can cover a congruent copy of every convex planar set of diameter 1.
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Mahler volume product constant
Let K⊂ℝ^n be a centrally symmetric convex body (compact, convex, with non-empty interior) satisfying K=-K. Its polar body is K^∘ := y∈ℝ^n: ⟨ x,y⟩ ≤ 1 for all x∈ K. The volume product of K is vp(K) := Vol_n(K) Vol_n(K^∘).
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Max to min ratios
Let n,d ≥ 2. Let C(d,n) denote the largest quantity such that, given any n distinct points x_1,…,x_n in R^d, the maximum distance max_1 ≤ i < j ≤ n ‖x_i-x_j‖ between the points is at least C(d,n) times the minimum distance min_1 ≤ i < j ≤ n ‖x_i-x_j‖. Establish upper and lower bounds for C(d,n).
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Moser's convex worm cover constant
C_13a is the infimal area of a convex domain Ω that can contain a rigid motion (translation + rotation; no reflections) of every planar arc (curve, or "worm") of length 1.
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Moving Sofa Constant
The moving sofa constant C_41a=A is the maximum area of a connected, rigid planar shape that can maneuver through an L-shaped corridor of unit width. The corridor is formed by two semi-infinite strips of width 1 meeting at a right angle.
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Packing in a dilate
For any n ≥ 1 and a geometric shape P (e.g. a polygon, a polytope or a sphere), let C(n, P) denote the smallest scale s such that one can place n identical copies of P with disjoint interiors inside another copy of P scaled up by a factor of s.
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Pairwise touching cylinders
Is it possible for seven infinite circular cylinders C_1,…,C_7 of unit radius to touch all the others?
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Points on sphere maximizing the volume
For any n ≥ 4, Let C(n) denote the maximum volume of a polyhedron with n vertices that all lie on the unit sphere S^2. What is C(n)? Which polyhedra attain the maximum volume?
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Reverse Brunn-Minkowski constant
For subsets K,L⊂ℝ^n, their Minkowski sum is K+L := x+y: x∈ K, y∈ L. In general, one cannot expect a reverse Brunn-Minkowski inequality for arbitrary compact sets, even with a fixed multiplicative constant.
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Smallest dimension in which Borsuk’s conjecture fails
For a bounded set X⊂ ℝ^n, its diameter is diam(X) := sup‖x-y‖_2: x,y∈ X. Let b(X) be the smallest integer m such that X can be written as a union X = X_1 ∪ ⋯ ∪ X_m with diam(X_i) < diam(X) for all i=1,…,m.
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Sphere packing density in ℝ^4
C_36=Δ_4 is the (optimal) sphere packing density in ℝ^4, i.e. the largest fraction of ℝ^4 that can be covered by congruent balls with disjoint interiors.
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Spherical Designs
A spherical t-design on the d-dimensional sphere S^d ⊂ R^d+1 is a finite set of points X ⊂ S^d such that for any polynomial P of degree at most t, the average value of P over X is equal to the average value of P over the entire sphere S^d.
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Tammes problem
For N ≥ 2, let C(N) denote the maximal value of the energy E(z_1,…,z_N) := min_1 ≤ i < j ≤ N ‖z_i-z_j‖ where z_1,…,z_N range over points in S^2. Establish upper and lower bounds on C(N) that are as strong as possible. What type of configurations z_1,…,z_N come close to achieving the maximal energy?
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The no 5 on a sphere problem
For n a natural number, let C(n) denote the size of the largest subset of [n]^3 = 1,…,n^3 such that no 5 points lie on a sphere or a plane. Obtain upper and lower bounds for C(n) that are as strong as possible.
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The Ovals Problem
Let C denote the infimal value of λ_0(γ), the least eigenvalue of the Schrödinger operator H_γ = -d^2/ds^2 + κ^2(s) associated with a simple closed convex curve γ parameterized by arclength and normalized to have length 2π, where κ(s) is the curvature.
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The three-dimensional moving sofa problem with two perpendicular turns
Define C to be the largest volume of a connected bounded subset S_3 of R^3 that can continuously pass through a three-dimensional snake-shaped corridor with a unit square cross-section, consisting of two turns in the x-y and y-z planes that are far apart. What is C?
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Tight alternating knot constant
C_22b = b_o is the largest constant for which one has an inequality L ≥ b_o C for all knots that admit an alternating diagram, where L is the ropelength of a knot (or link) with crossing number) C.
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Tight knot constant
C_22a is the largest constant for which one has an inequality L≥ C_22aC^3/4 for all knots, where L is the ropelength of a knot (or link) with crossing number) C.
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Ben Green's Open Problem 41
How many rotated (about the origin) copies of the 'pyjama set' \(x, y) ∈ ℝ^2 : dist(x, ℤ) ≤ ε\ are needed to cover ℝ^2? That is, determine the minimal number of rotations as a function of ε > 0.
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Borsuk's conjecture
Borsuk's conjecture in dimension 4, the smallest open case.
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Conjecture about cardinality of Lindelöf spaces
Is there a Lindelöf Tychonoff space with singletons as Gδ sets with cardinality greater than the continuum? Note: the cited paper uses a blanket convention that all spaces are Tychonoff.
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Conjectures about Weakly First Countable spaces
Problem 2 in [Ar2013]: Give an example in ZFC of a weakly first- countable compact Hausdorff space X such that 𝔠 < |X|. Note: [Ar2013] uses a blanket convention that all spaces are Tychonoff and "compact" means compact Hausdorff.
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Conjectures around homogeneous topological spaces
Problem 13 in [Ar2013]: Is it true that every infinite homogeneous compact hausdorff space contains a non-trivial convergent sequence?
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Erdős Problem #100
Is the diameter of A at least Cn for some constant C > 0?
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Erdős Problem #101
Given n points in ℝ^2, no five of which are on a line, the number of lines containing four points is o(n^2).
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Erdős Problem #102
Let c > 0 and let h_c(n) be such that for any n points in ℝ^2 with at least cn^2 lines that each contain more than three of the points, some line contains h_c(n) of the points. Is it true that, for fixed c > 0, h_c(n) → ∞?
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Erdős Problem #103
Let h(n) count the number of incongruent sets of n points in ℝ^2 which minimise the diameter subject to the constraint that d(x,y)≥ 1 for all points x≠ y. Is it true that h(n)→ ∞?
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Erdős Problem #104
Given n points in ℝ^2 the number of distinct unit circles containing at least three points is o(n^2).
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Erdős Problem #1070
Let f(n) be maximal such that, given any n points in ℝ^2, there exist f(n) points such that no two are distance 1 apart. Estimate f(n). In particular, is it true that f(n)≥ n/4? This is the first part, a problem of Erdős [Er87b, p. 171]. Since f(n)=Θ(n) is already known, we ask for lim f(n)/n.
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Erdős Problem #1082
Let A⊂ ℝ^2 be a set of n points with no three on a line. Does A determine at least ⌊ n/2⌋ distinct distances?
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Erdős Problem #1083
Let d≥ 3, and let f_d(n) be the minimal m such that every set of n points in ℝ^d determines at least m distinct distances. Estimate f_d(n) - in particular, is it true that f_d(n)=n^2/d-o(1)?
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Erdős Problem #1086
Let g(n) be minimal such that any set of n points in ℝ^2 contains the vertices of at most g(n) many triangles with the same area. Estimate g(n). Only triangles of positive area are counted.
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Erdős Problem #1087
Let f(n) be minimal such that every set of n points in ℝ^2 contains at most f(n) many sets of four points which are 'degenerate' in the sense that some pair are the same distance apart. Estimate f(n) - in particular, is it true that f(n)≤ n^3+o(1)?
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Erdős Problem #1088
Let f_d(n) be the minimal m such that any set of m points in ℝ^d contains a set of n points for which any two determined distances are distinct. Erdős Problem 1088 asks to estimate f_d(n). In particular, is it true that, for every fixed n ≥ 3, f_d(n) = 2^o(d) as d → ∞?
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Erdős Problem #1207
Let P_d(n) be such that in any set of n points in ℝ^d there exist at least P_d(n) many points which do not contain an isosceles triangle. Estimate P_d(n) - in particular, is it true that P_2(n)<n^1-c for some constant c>0?
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Erdős Problem #173
In any 2-colouring of ℝ^2, for all but at most one triangle T (up to congruence), there is a monochromatic congruent copy of T. The colouring is arbitrary. No measurability is assumed. ≅ compares vertices with the same label, so the conclusion allows a relabelling σ.
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Erdős Problem #174
A finite set A⊂ ℝ^n is called Ramsey if, for any k≥ 1, there exists some d=d(A,k) such that in any k-colouring of ℝ^d there exists a monochromatic copy of A. Characterise the Ramsey sets in ℝ^n. A copy of A is a set congruent to A, as in [EGMRSS73, p. 344].
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Erdős Problem #188
What is the smallest k such that ℝ^2 can be red/blue coloured with no pair of red points unit distance apart, and no k-term arithmetic progression of blue points with distance 1?
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Erdős Problem #212
Is there a dense subset of ℝ^2 such that all pairwise distances are rational?
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Erdős Problem #213
Let n ≥ 4. Are there n points in ℝ^2, no three on a line and no four on a circle, such that all pairwise distances are integers?
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Erdős Problem #217
For which n do there exist n points in ℝ^2, no three on a line and no four on a circle, that determine n-1 distinct distances whose multiplicities are 1,2,…,n-1? The condition n>0 excludes the empty configuration, since the requested number n-1 of distances is nonnegative.
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Erdős Problem #352
Is there some c > 0 such that every measurable A ⊆ ℝ^2 of measure ≥ c contains the vertices of a triangle of area 1?
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Erdős Problem #503
What is the size of the largest A ⊆ ℝ^n such that every three points from A determine an isosceles triangle? That is, for any three points x, y, z from A, at least two of the distances |x - y|, |y - z|, |x - z| are equal.
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Erdős Problem #506
Erdős Problem #506
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Erdős Problem #507
Let α(n) be such that every set of n points in the unit disk contains three points which determine a triangle of area at most α(n). Estimate α(n).
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Erdős Problem #508
The Hadwiger–Nelson problem asks: How many colors are required to color the plane such that no two points at distance 1 from each other have the same color? It is known that 6 ≤ χ(ℝ^2) ≤ 7; see HadwigerNelsonAtLeastSix and HadwigerNelsonAtMostSeven.
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Erdős Problem #588
Let f_k(n) be minimal such that if n points in ℝ^2 have no k+1 points on a line then there must be at most f_k(n) many lines containing at least k points. Is it true that f_k(n) = o(n^2) for k ≥ 4? The case k = 4 is Erdős Problem 101.
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Erdős Problem #653
Let x_1,…,x_n∈ ℝ^2 and let R(x_i)=\# lvert x_j-x_irvert : j≠ i, where the points are ordered such that R(x_1)≤ ⋯ ≤ R(x_n). Let g(n) be the maximum number of distinct values the R(x_i) can take. Is it true that g(n) ≥ (1-o(1))n?
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Erdős Problem #654
Let f(n) be as in erdos_654.parts.i. Is it true that f(n) > (1/3+c)n for some constant c>0 and all large n?
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Erdős Problem #657
Is it true that if A⊂ ℝ^2 is a set of n points such that every subset of 3 points determines 3 distinct distances (that is, A has no isosceles triangles, degenerate ones included) then A determines at least f(n)n distinct distances, for some function f with f(n)→ ∞?
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Erdős Problem #660
Let x_1, …, x_n ∈ ℝ^3 be the vertices of a convex polyhedron. Are there at least (1 - o(1)) n/2 many distinct distances between the x_i?
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Erdős Problem #661
Are there, for all large n, some points x_1,…,x_n,y_1,…,y_n∈ ℝ^2 such that the number of distinct distances d(x_i,y_j) is o(n/√(log n))? We require all 2n points to be distinct; without any distinctness one could take all points equal and get a single distance.
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Erdős Problem #668
Is it true that the number of incongruent sets of n points in ℝ^2 which maximise the number of unit distances tends to infinity as n→∞? The actual maximal number of unit distances is the subject of Erdős Problem 90.
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Erdős Problem #669
Let F_k(n) be minimal such that for any n points in ℝ^2 there exist at most F_k(n) many distinct lines passing through at least k of the points, and f_k(n) similarly but with lines passing through exactly k points.
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Erdős Problem #670
Let A⊆ ℝ^d be a set of n points such that all pairwise distances differ by at least 1. Is the diameter of A at least (1+o(1))n^2? The quantifiers in the source are ambiguous.
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Erdős Problem #757
What is the supremum of the set of admissible numbers?
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Erdős Problem #827
Let n_k be minimal such that if n_k points in ℝ^2 are in general position then there exists a subset of k points such that all C(k, 3) triples determine circles of different radii. Determine n_k.
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Erdős Problem #831
Let h(n) be maximal such that in any n points in ℝ^2 (with no three on a line and no four on a circle) there are at least h(n) many circles of different radii passing through three points. Estimate h(n). Erdős asks this in [Er75h, pp. 2-3] (see also [Er92e]); see also Problems 104 and 506.
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Erdős Problem #838
Let f(n) be maximal such that any n points in ℝ^2, with no three on a line, determine at least f(n) different convex subsets. Estimate f(n) - in particular, does there exist a constant c such that lim log f(n)/(log n)^2=c? A question of Erdős and Hammer [Er78c, pp. 52-53]. This is the first part.
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Erdős Problem #89
Erdős [Er46] asked whether every set of n distinct points in ℝ^2 determines ≫ n/√(log n) many distinct distances.
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Erdős Problem #91
Suppose A⊂ ℝ^2 has lvert Arvert=n and minimises the number of distinct distances between points in A. Prove that for large n there are at least two (and probably many) such A which are non-similar.
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Erdős Problem #959
Let A⊆ ℝ^2 be a set of size n and let d_1,…,d_k be the set of distinct distances determined by A. Let f(d) be the number of times the distance d is determined, ordered so that f(d_1)≥ f(d_2)≥ ⋯ ≥ f(d_k).
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Erdős Problem #96
If n points in ℝ^2 form a convex polygon then there are O(n) many pairs which are distance 1 apart.
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Erdős Problem #97
Does every convex polygon have a vertex with no other 4 vertices equidistant from it?
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Erdős Problem #98
Let h(n) be such that any n points in ℝ^2, with no three on a line and no four on a circle, determine at least h(n) distinct distances. Does h(n)/n→ ∞?
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Erdős Problem #982
If n distinct points in ℝ^2 form a convex polygon then some vertex has at least lfloorn/2⌋ different distances to other vertices.
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Erdős Problem #99
For sufficiently large n, is it the case that any set of n points with minimum distance 1 that minimizes diameter must contain an equilateral triangle of side length 1?
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Green's Open Problem 42
Can the Cohn-Elkies scheme be used to prove the optimal bound for circle-packings in 2 dimensions?
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Inscribed square problem
Inscribed square problem Does every Jordan curve admit an inscribed square?
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Integer part of area of a regular polygon with n sides each of length 1
"Usually (perhaps always?) ⌊ n^2 / (4π) - π / 12 ⌋ for a polygon of circumference n. Note that the area of a circle with circumference C is C^2 / (4π)."
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Mathoverflow 34145
Can a unit square be covered by rectangles of width 1 / (n + 1) and height 1 / (n + 2)?
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Moser's Worm
Moser's Worm Problem What is the minimal area (or greatest lower bound on the area) of a shape that can cover every unit-length curve?
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Packing
What is the smallest square that can contain 11 unit squares? Reference: Wikipedia
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The Bing-Borsuk Conjecture
The Bing-Borsuk Conjecture: every n-dimensional homogeneous absolute neighborhood retract is a topological n-manifold. A topological space X is an n-dimensional manifold when T2Space X ∧ Nonempty (ChartedSpace (Fin n → ℝ) X).
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The Erdős–Ulam problem
The Erdős–Ulam problem (Ulam 1945). Is there a dense subset of the plane whose points are at pairwise rational distances? Conjecturally, no such set exists.
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The first Atiyah–Sutcliffe conjecture
Atiyah–Sutcliffe Conjecture 1, stated as Conjecture 1.1 in Mazur–Petrenko: the configuration polynomials are linearly independent.
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Toronto spaces
Any T2, Toronto space is discrete.
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