Problems
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For A ⊆ F_p let A^* = (A+A) ∪ (AA). Sárközy conjectured that for all large primes, every set of size at least c√p has A^* = F_p-like covering behaviour. Disproved with an explicit construction from the classical cross-ratio orbit, together…
Every polynomial map C^n → C^n with constant nonzero Jacobian determinant is invertible, with a polynomial inverse.
The Smith-Ward theorem realizes the first k essential matrix ranges of an operator as the matrix ranges of a compact perturbation. The classical Smith-Ward problem asks whether that perturbation can be chosen independently of k,…
Teschner conjectured that every finite simple graph G with at least one edge satisfies b(G) ≤ 3/2Δ(G), where b(G) is the bondage number and Δ(G) is the maximum degree. Yavari gives a connected cubic bipartite graph on 18 vertices with…
Around the Kemeny median problem, which stays open for m=3 and m=5 voters, the paper refutes three conjectures on tournament inducibility: both conjectures of Milosz, Hamel and Pierrot (the 3-cycle extension for odd m≥5, and FAS=HS_3 at…
For the Erdős–Pomerance functions F(n) and h_P(n) counting how many consecutive integers are needed to contain a distinct multiple of each integer, respectively prime, up to n, the paper proves F(n) ≥ h_P(n) ≥ nexp((log 2/2 - o(1))log…
Is the chromatic symmetric function X_G Schur positive for every claw-free graph G? Two explicit 12-vertex line graphs have Schur coefficients -64 and -40 at s_(3,3,3,3).
Kinoshita conjectured that every embedded projective plane in S^4 is reducible. False: an irreducible embedded projective plane exists in S^4. The construction also answers both parts of Problem 4.37 of the Kirby problem list.
Can a Cohn-Elkies auxiliary function certify the best known sphere packing in dimension 36 as optimal? No. An explicit dual-feasible point for the Cohn-Elkies linear program, built from weight-18 modular forms for Γ_0(24), shows the…
For single-source unsplittable flow, every fractional flow can be rounded to an unsplittable flow whose cost is no higher than the fractional cost, while each arc's load is exceeded by at most the maximum demand. (The cost version of…
VibeMathed records no statement for this problem. See erdosproblems.com for the original.
Let μ be a probability measure on the unit circle with Verblunsky coefficients α. Lukic conjectured that a weighted entropy condition with finitely many critical points is equivalent to a decomposition of α into components localized at…
Steurer conjectured in 2010 that any family of n unit vectors with polynomially small average correlation E_i,j|⟨ v_i,v_j⟩| ≤ n^-ε contains linear-sized constant-separated sets. Refuted in a strong sense, using sparse high-dimensional…
Erdos and Graham asked whether a positive-density subset of 1,…,N can avoid having any two distinct elements a,b whose unit fractions average to a unit fraction. It can: there is a constant c>0 such that for all large N some A ⊆ 1,…,N of…
The unrestricted planar Berenstein conjecture holds that overdetermined Dirichlet-Neumann data characterize the disc. Disproved: a bounded simply connected domain with real-analytic Jordan boundary that is not a disc, carrying a nonzero…
Ziegler proved every simplicial d-dimensional 0/1-polytope has at most 2d vertices, and asked whether attaining 2d vertices forces central symmetry (i.e. a 0/1 cross-polytope). Known true for d ≤ 6; open since ~2000.
Lions asked whether the variational solution of a non-autonomous divergence-form problem has maximal L2-regularity under Holder continuity in time of the coefficients. Disproved at the half-Holder endpoint: a bounded, uniformly elliptic,…
Mauri and Moraga posed a two-part question about log Calabi-Yau pairs whose boundary decomposes into big divisors. Both parts have negative answers.
Does every lattice of density above one admit a Gabor frame with a nice window? No. For every dimension d > 1 there are explicit criteria on lattices Λ ⊂ R^2d with D(Λ) > 1 such that no function with continuous Zak transform generates a…
The paper exhibits an explicit integer polynomial in five variables, of total degree 14 with constant Hessian determinant 128, whose gradient is not injective. Its formal Legendre transform is therefore not a polynomial, so the Hessian…