Each problem states how progress is verified and what counts as a contribution. Besides the problems curated here, the catalogue includes open conjectures from Formal Conjectures (with Lean statements), optimization constants and the AlphaEvolve problems. Know one that belongs here? Propose a problem.
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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).
Determine τ5, the maximum number of non-overlapping unit spheres touching a central unit sphere in R^5. Currently 40 ≤ τ5 ≤ 44.
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.
Borsuk's conjecture, open range: every bounded subset of ℝ^n with at least two points can be partitioned into n + 1 sets of strictly smaller diameter, for 4 ≤ n ≤ 62. The conjecture is known to be true for n ≤ 3 and false for n ≥ 63.
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.
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.
Problem 13 in [Ar2013]: Is it true that every infinite homogeneous compact hausdorff space contains a non-trivial convergent sequence?
Is the diameter of A at least Cn for some constant C > 0?
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).
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) → ∞?
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)→ ∞?
Given n points in ℝ^2 the number of distinct unit circles containing at least three points is o(n^2).
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?
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)?
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 → ∞?
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?
Is there a dense subset of ℝ^2 such that all pairwise distances are rational?
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?
Is there some c > 0 such that every measurable A ⊆ ℝ^2 of measure ≥ c contains the vertices of a triangle of area 1?
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.
Erdős Problem #506
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).
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?
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?
Erdős [Er46] asked whether every set of n distinct points in ℝ^2 determines ≫ n/√(log n) many distinct distances.
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.
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).
If n points in ℝ^2 form a convex polygon then there are O(n) many pairs which are distance 1 apart.
Does every convex polygon have a vertex with no other 4 vertices equidistant from it?
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→ ∞?
If n distinct points in ℝ^2 form a convex polygon then some vertex has at least lfloorn/2⌋ different distances to other vertices.
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?
Can the Cohn-Elkies scheme be used to prove the optimal bound for circle-packings in 2 dimensions?
Inscribed square problem Does every Jordan curve admit an inscribed square?
"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)."
Can a unit square be covered by rectangles of width 1 / (n + 1) and height 1 / (n + 2)?
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?
What is the smallest square that can contain 11 unit squares? Reference: Wikipedia
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).
Atiyah–Sutcliffe Conjecture 1, stated as Conjecture 1.1 in Mazur–Petrenko: the configuration polynomials are linearly independent.
Any T2, Toronto space is discrete.
