Problems
No person has reviewed any of this; every judgement here is a machine's.
A cyclic coloration of a triangulation of a closed 2-manifold gives the faces around every vertex distinct colors. Chen and Lawrencenko made two conjectures about the cyclic chromatic number of minimal triangulations in 1999. Their second…
Is the O'Shea-Zames-Falb multiplier test necessary for robust stability of Lur'e systems with slope-restricted nonlinearities, as conjectured by Carrasco? No: there is a stable Lur'e interconnection, certified by a full-block multiplier,…
Casalaina-Martin and Zhjeqi proved that the first Chern class of every torsion-free coherent quotient of a tensor power of the logarithmic cotangent sheaf is pseudo-effective, noting in Remark 4.5 that torsion-freeness was imposed only for…
Pach conjectured that n Jordan arcs, pairwise crossing exactly once with no triple points, have O(n) tangent pairs. The best known bound stood at O(n^7/4); the paper improves it to O(n^3/2) (and O(n^5/3) in the at-most-one-crossing…
For a graph G, its vertex deck is the multiset of graphs obtained by deleting one vertex. Bowler, Brown, and Fenner (BBF) proposed 2⌊(n−1)/3⌋ as the maximum possible overlap between the decks of two nonisomorphic n-vertex graphs, for all…
Pozdnyakov, Willatt, Bartók, Ortner, Csányi and Ceriotti showed in 2020 that the 2-, 3- and 4-point correlations of an atomic neighbour density are incomplete: noncongruent environments can share them exactly. Every degeneracy found since…
How dense can a sum-free subset of the lattice cube 1,…,n^d be? Aydinian and Cameron asked for the limiting density, which is also Problem 6 in Ben Green's list of 100 open problems. The natural conjecture is that the optimum is a slice x…
Does the block 11 occur infinitely often in the base-2 expansion of the Erdős-Borwein constant E = ∑_n ≥ 1 1/2^n - 1? Posed by Crandall in 2012.
In the Frankl-Pach-Erdős circle of VC-dimension problems, the first arXiv version of the paper posed the k=3 case of a witness construction question. ChatGPT 5.4 Pro answered it; the published construction generalizes the model's response,…
The group version of the Matrix Spencer conjecture holds: for every finite group G there are signs ε ∈ ± 1^G with |∑_g ∈ G ε_g ρ(g)| ≤ C√|G|, where ρ is the left regular representation and C is universal.
Whether negative-partial-transpose states undistillable from one copy become distillable from finitely many copies is a basic open problem in entanglement theory. In the canonical two-parameter DiVincenzo family used as its…
For an unkilled Levy process ξ drifting to +∞ with all positive exponential moments, let I_ξ = ∫_0^∞ e^-ξ_t dt and X_ξ = 1/I_ξ. Bertoin and Yor proved X_ξ is moment-determinate when ξ has no positive jumps and conjectured that this…
For a multidimensional reflected diffusion, does the basic adjoint relationship uniquely characterize the stationary distribution? The question had stood unresolved for more than thirty-five years since the BAR approach was introduced. For…
Can any single-pass semi-streaming algorithm beat the naive greedy 1/2-approximation for maximum matching? No. No single-pass semi-streaming algorithm, deterministic or randomized, achieves a better-than-half approximation, so greedy is…
The dimer constant of Z^3, the exponential growth rate of perfect matchings of the cubic lattice, has no closed form and is pinned only by bounds. The upper bound improves from Lundow's 0.457547, standing since 2001, to 0.452130, via…
Deng, Tidor and Zhao asked whether [N] admits a coloring with N^o(1) colors and no symmetrically coloured 4-term arithmetic progression, giving an O(N^log_223) coloring. The paper gives an O_k(N^4/k^2) coloring of [N] avoiding…
What is the optimal competitive ratio for online vertex cover when edges arrive one at a time? The paper proves a tight factor-2 lower bound via a reduction in the blueprint framework of Assadi, Jiang and Xiang, closing the gap left by…
Let U_n⊂ R^n be the difference polytope of a regular n-simplex such that U_n circumscribes a sphere of diameter 1. Then every set of diameter 1 in R^n is covered by a rotated copy of U_n.
Sharp global and almost-everywhere convergence rates for periodic homogenization of viscous quadratic Hamilton-Jacobi equations, settling the sharpness question left open by the first-order theory.
Reiner conjectured a description of the homotopy types of intervals in higher Bruhat orders. In corank 3 it holds: the facial intervals of B(n,n-3) are exactly the spherical intervals, and every other interval is contractible.