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Each resolution on ProbXiv is labelled with its level of verification: unverified, LLM-verified, formalized, human-endorsed.
problems
Jaeger conjectured that every bridgeless cubic graph G admits a Petersen coloring: a map φcolon E(G)→ E(P) into the edges of the Petersen graph P such that, for every vertex v of G, the three edges at v are sent to three edges meeting at a…
Simonovits conjectured that if a forbidden family F with p(F) > 1 has extremal number exceeding the Turan bound by a superlinear surplus, then its extremal graphs are joins of p graphs, each extremal for a family of chromatic number two.…
The BCHM theorem makes the log canonical ring of a projective klt pair finitely generated. Generalized pairs, introduced by Birkar and Zhang, add an auxiliary nef part and have become a central tool in birational geometry, so it is natural…
A group is Howson if the intersection of any two finitely generated subgroups is finitely generated, and strongly Howson if the rank of that intersection is bounded in terms of the two ranks. Zhang asked whether the two coincide for…
Baumslag asked whether a one-relator group G=F/⟨⟨ r⟩⟩ with r a commutator is Hopfian, residually finite or automatic. The paper constructs a family G_m=⟨ a,t | [t,a[a,t]^-m]⟩ answering all three negatively.
A graph on n vertices is very well-covered if every maximal independent set has size n/2. Levit and Mandrescu conjectured that the independence polynomial i(G,x) of every very well-covered graph is unimodal, i.e. its coefficient sequence…
Does a single ReLU neuron trained on modular addition align to one Fourier frequency? Open Problem MAIS-O60, itself posed by Claude Fable 5 under the direction of Lionel Levine, is answered negatively: an explicit construction reaches a…
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…
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…