Problems

For D_3(m) = vecC_m square vecC_m square vecC_m, can the full arc set be partitioned into three directed Hamilton cycles for every integer m ≥ 3?

Among sufficiently large one-separated planar point sets, does the triangular lattice maximize the number of distances below each threshold? Explicit rational oblique lattices beat the triangular lattice under several closed- and…

For a pure O-sequence h = (h_0, …, h_e) of codimension three and type two, is h_i^2 ≥ h_i-1 h_i+1 for every interior index i? The stated monomial case is proved; the broader level-Hilbert-function case remains open.

For every finite connected simple graph G, is the order of the largest induced tree at least girth(G) - 1 + ecc(G, center(G)), where the last term is the eccentricity of the centre set? Answered affirmatively, with a Lean proof.

Can there be a finite covering system of the integers with distinct moduli, all of which are odd and greater than 1?

For every finite connected graph, is girth(G) + 1 at most the product of its largest induced-tree order and its second-smallest degree?

Given n and 1 ≤ c ≤ n!, can n distinct group elements be chosen so that their n! ordered products take exactly c distinct values? Constructions realize every c.

Do arbitrarily large 4-chromatic edge-critical graphs exist with minimum degree bounded below by a positive constant times the number of vertices?

Let A ⊂ N be infinite with no distinct a, b, c ∈ A such that a | (b + c) with b, c > a. Can |A ∩ [1, N]|/√N have positive lower limit? Must every such A fall below N^1-c infinitely often?

VibeMathed records no statement for this problem. See erdosproblems.com for the original.

VibeMathed records no statement for this problem. See erdosproblems.com for the original.

Does the Hodge bundle Ω_g over the moduli stack of genus g ≥ 2 curves contain any nontrivial sub-bundles? Posed by Dawei Chen around 2015; the answer is no.

The swap chain flips checkerboard 2×2 blocks to sample 0/1 matrices with fixed row and column sums. Kannan, Tetali and Vempala conjectured in 1997 that it mixes in polynomial time for all feasible margins; the lazy chain is shown to have…

Does there exist A=a_1<a_2<…⊂ N which is a minimal basis of order 2 (every large integer is the sum of 2 elements from A, and no proper subset of A has this property) such that lim_k→ ∞a_k/k^2=c for some c≠ 0? A claimed construction gives…

Ji, Li and Wang conjectured in 2024 that every parallel chip-firing game on a finite connected graph whose chip count lies strictly between 2|E|-|V| and 2|E| has period exactly 2, generalizing the middle rung of Levine's devil's staircase…

If A(x) counts integers satisfying the Sylow divisor condition, determine the constant c in A(x)/x = exp(-(c + o(1)) √log x loglog x). The claimed exact value is c = 1/(2√log 2).

For |A| = n, how small can the cofactor set Q(A) = a / gcd(a,b) : a, b ∈ A be? The answer is h(n) = n^1/2 + o(1): a new upper bound h(n) ≤ n^1/2 exp(O(√log n)) matches the classical lower bound.

What is the maximum volume of a convex body in R^n whose centroid is its only interior lattice point? Ehrhart conjectured the extremal value in 1964; the sharp maximum is now determined in every dimension.

VibeMathed records no statement for this problem. See erdosproblems.com for the original.

Borsuk's conjecture asked whether every bounded set in R^n can be partitioned into n+1 subsets of smaller diameter. It is false in dimension 63: there is a set of 321 points in R^63 whose smaller-diameter subsets have at most 5 points, so…