Problems
Everything in the archive: the problem as it was posed, what has been attempted against it, and who checked each attempt. The mark down the left of the list says who has looked — a person, a machine, or nobody yet. Human reviews and machine checks are counted separately and are never added together.
266 problems
A question of Averkov, Hofscheier and Nill on whether the Ehrhart h^*-polynomial of a lattice polytope of large lattice width is real-rooted. Proved in fixed dimension for sufficiently large lattice width, giving strict log-concavity and…
What is the largest A⊆1,…,N such that all subset sums ∑_n∈ S1/n (over S⊆ A) are distinct?
A convex body is in Faber-Krahn position if it minimizes the first Dirichlet eigenvalue within its volume-preserving linear orbit. The paper proves this position is unique up to orthogonal transformations, answering a question of…
Does there exist a group with more than one but only finitely many maximal locally soluble normal subgroups? An explicit group with exactly two settles it.
How many pairwise non-overlapping infinite circular cylinders of unit radius can simultaneously touch a unit ball? Kuperberg conjectured in 1990 that the maximum is six.
For every finite set A⊂ Z with |A|≥ 2, define C(A)=log(|A+A|/|A|)/log(|A-A|/|A|). Determine the largest possible value of C(A), equivalently the least universal exponent c such that |A+A|/|A| ≤ (|A-A|/|A|)^c for every such set A. The…
Zhu, Gyori, He, Lv, Salia and Xiao conjectured the maximum number of copies of a fixed cycle in an n-vertex graph of bounded circumference, attained by the join of a clique with an independent set. For every fixed s ≥ 3 and L ≥ 2s+2 and…
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.
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.
Assuming the Unique Games Conjecture, it is NP-hard to approximate MAX-3-CUT better than the Frieze-Jerrum semidefinite program does, and similarly for Quantum MAX-CUT: the sharpness question in the Khot-Kindler-Mossel-O'Donnell line,…
Krauth and Mezard predicted in 1989 that the storage capacity of the Ising perceptron at zero margin is an explicit constant α_⋆ ≈ 0.8330786. Ding and Sun proved the matching lower bound and Huang the upper bound, but each was conditional…
Under smoothness, positivity, decay and score assumptions, are all steady solutions of the Coulomb Vlasov-Maxwell-Landau system on T^3 × R^3 necessarily spatially uniform Maxwellians?
Is computing a Kemeny-optimal aggregate ranking NP-hard when the input consists of exactly three complete rankings? Hardness was known for every even n ≥ 4; three voters was the minimal open case, and n = 2 is polynomial-time solvable.
Talagrand's convexity problem asks whether a universal number of Minkowski sum operations turns any set of large Gaussian measure into one containing a convex body of comparable measure. It is equivalent to a question about subgaussian…
For a perfect field k and a representation-infinite finite-dimensional k-algebra A, the Auslander–Reiten quiver of A has infinitely many connected components. This establishes a conjecture of Auslander, Reiten and Smalø, for…
Wu and Santhanam asked whether one can determine, from an increasing i.i.d. sample of binary random matrices, whether the unknown mean matrix is diagonalizable, while making only finitely many errors almost surely. Answered affirmatively…
VibeMathed records no statement for this problem. See erdosproblems.com for the original.
For the switch-walk-switch lamplighter walk on Z_2 wr T_d, prove the sharp asymptotic p_2n(e,e) = ρ_d^2n exp[-(π^2 (log(d-1))^2 + o(1)) n/log^2 n] with ρ_d = frac2√d-1d.