Problems
No person has reviewed any of this; every judgement here is a machine's.
Do a finite group's order together with ∑_g ∈ G φ(|g|) determine whether the group is simple? A simple and a non-simple group of order 6048 share the statistic 23984.
Banach asked in 1932 whether a real Banach space X whose ndimensional subspaces, for some fixed 1 < n < dim X, are all isometric must be a Hilbert space. Gromov proved the conjecture for even n, and subsequent work settled several…
How few vertices can a triangulation of RP^5 have? The paper presents a 6-dimensional centrally symmetric simplicial polytope whose antipodal boundary quotient gives a 24-vertex triangulation, far below previous constructions in the…
If A generates a bounded C_0-semigroup on a Hilbert space and has dense range, does A^-1 also generate a bounded C_0-semigroup? Posed by deLaubenfels in 1988. Answered negatively: Lorist, Meyries and Veraar construct a bounded operator…
Let k≥ 3 and f_k(N) be the maximum of ∑_n∈ A1/n over all A⊆1,…,N containing no k subsets with the same pairwise least common multiple. Estimate f_k(N). The claimed answer: f_k(N)=(log N)^γ_k+o(1), where γ_k is a weighted generalization of…
Let T_k be the least t such that every equinumerous t-coloring of [tn] contains a rainbow k-term arithmetic progression. Jungic, Licht, Mahdian, Nesetril and Radoicic conjectured T_k = Θ(k^2); Conlon, Fox and Sudakov proved T_k = O(k^2 log…
Akbari, Alikhani, Oboudi and Peng conjectured in 2010 that 0 and -2 are the only integer roots of the domination polynomial D(G, x), proven for trees and unicyclic graphs and verified exhaustively for small orders. The paper gives a…
For a sequence of n distinct reals, determine the largest constant c such that some monotonic subsequence always has sum exceeding (c-o(1))·(1/√n) times the total sum. Resolved as c = 1.
Seymour conjectured that every oriented graph has a vertex x with |N^++(x)| ≥ |N^+(x)|. It holds for oriented graphs of minimum out-degree exactly 7, the first improvement to the out-degree threshold since Kaneko and Locke settled degree 6…
Online Shadow Tomography with log m dependence, while retaining poly(log(d)/ε) dependence. Also, matching the best classical bounds for Adaptive Data Analysis
If CT_(k) is generated by all horizontal class transpositions with modulus at most k, is CT_(k) ≅ S_lcm(2,…,k) for every k ≥ 4?
Crouzeix conjectured in 2004 that for every square complex matrix A and every polynomial p, lVert p(A)rVert ≤ 2 max_z ∈ W(A) |p(z)|, where W(A) is the numerical range of A - that is, the numerical range is a 2-spectral set. Crouzeix proved…
Must the right-relatively convex subgroups of a right-orderable nonabelian group form a sublattice of its subgroup lattice? A construction shows they need not.
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.
How well separated can a family of point-line pairs in the unit square be? For every ε > 0 there are arbitrarily large families (x_1,ℓ_1),…,(x_n,ℓ_n) in [0,1]^2 with x_i ∈ ℓ_i and dist(x_i,ℓ_j) ≥ n^-2/3-ε for all i ≠ j. Combined with…
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.
The interchange graph G(R,S) has the (0,1)-matrices with row sums R and column sums S as vertices, adjacent when they differ by a single 2× 2 interchange. Brualdi asked whether G(R,S) is always Hamiltonian. It satisfies more: it is…
For the Fubini numbers a(n), is a(n) = ∑_k=0^2^n-1-1 A284005(k) for every n > 0, as conjectured on the OEIS in 2018?