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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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.
