OpenAI reports the exact cycle–clique Ramsey numbers
- Mathematics
For all integers m≥n≥3 except (m,n)=(3,3), R(C_m,K_n)=(m−1)(n−1)+1; the exceptional value is R(C₃,K₃)=6.
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1033 matching entries · Page 17 of 21
OpenAI reports a counterexample to unrestricted infinite matroid packing and covering
- Mathematics
In ZFC, two self-dual partitional matroids on a countably infinite common ground set admit neither a packing/covering partition nor an intersection witness.
OpenAI reports a power saving for planar halving lines
- Mathematics
Some absolute ε>0 and C bound the number of unordered halving pairs in every sufficiently large even n-point planar set with no three collinear by Cn^(4/3−ε).
OpenAI reports balanced counterexamples to Ryser’s covering conjecture
- Mathematics
For every sufficiently large prime q, a finite intersecting (q+1)-partite (q+1)-uniform hypergraph has covering number q+1 and exactly q+1 nonisolated vertices in each part.
OpenAI reports the entropy-rate dimension formula for self-similar measures
- Mathematics
For every finite real self-similar probability measure with positive weights and nonzero contraction ratios of absolute value below one, dim_H μ=min(1,h_RW/χ), even with exact overlaps and unequal signed ratios.
OpenAI reports a zero dimension exponent for well-conditioned log-concave sampling queries
- Mathematics
For C² potentials with known minimizer at zero and I≤∇²V≤2I, exact value-and-gradient oracle sampling to total-variation error at most 1/10 uses at most Cε d^ε queries for every ε>0, while every algorithm needs at least c log d queries eventually.
OpenAI reports a two-fifths-power space simulation for one-tape machines
- Theoretical Computer Science
A fixed deterministic machine with one writable tape can be simulated through a supplied time cap T≥2 in O(T^(2/5) polylog(T+2)) work-space bits while preserving its finite-control and halting outcome.
OpenAI reports a depth-five lower bound for iterated matrix multiplication
- Theoretical Computer Science
Over every characteristic-zero field, for all sufficiently large n every syntactically homogeneous depth-five ΣΠΣΠΣ circuit computing the (1,1) entry of n independent n×n matrix products has at least n^(√n/400) gates.
OpenAI reports exponential state cost for two-way NFA complementation
- Theoretical Computer Science
For every n≥4, an explicit n-state two-way nondeterministic finite automaton has a complement requiring at least ½·2^⌊(n−4)/127⌋−1 states in any two-way nondeterministic finite automaton.
OpenAI reports the asymptotic average-sensitivity bound for polynomial thresholds
- Theoretical Computer Science
Every degree-at-most-d real polynomial threshold function on the uniform n-dimensional Boolean cube has average sensitivity at most 8d√n for 1≤d≤n.
OpenAI reports exact uniform sampling of contingency tables with arbitrary margins
- Mathematics
- Theoretical Computer Science
For any nonnegative row and column margins with equal totals, an exact uniform sampler for integer contingency tables terminates almost surely and has expected bit complexity polynomial in both dimensions and the binary input length.
OpenAI reports a cylinder-covering counterexample for a regular tetrahedron
- Mathematics
A regular tetrahedron has a finite cylinder cover with compact triangular perpendicular bases whose total area is strictly less than half its minimum orthogonal projection area.
OpenAI reports uniqueness in the smooth isotropic elasticity inverse problem
- Mathematics
On any bounded connected smooth domain in three-dimensional Euclidean space, equality of full static displacement-to-traction maps for two smooth real Lamé-modulus pairs with μ>0 and 3λ+2μ>0 on the closure forces both moduli to agree throughout the domain.
OpenAI reports the subcritical Hénon–Lane–Emden theorem
- Mathematics
For n≥2, p,q>0 and real weights A,B satisfying (n+A)/(p+1)+(n+B)/(q+1)>n−2, the Hénon–Lane–Emden system has no strictly positive entire solution continuous at the origin and C² away from it.
OpenAI reports convex tangent injectivity domains under weak MTW
- Mathematics
On every compact connected smooth Riemannian manifold of dimension at least two satisfying weak Ma–Trudinger–Wang curvature, each open tangent injectivity domain is convex.
OpenAI reports the cubic three-torus isoperimetric profile
- Mathematics
In the unit cubic flat three-torus, every volume-minimizing region is a ball, a circular tube about a shortest closed geodesic, a coordinate slab or the complement of one of these, in the predicted volume ranges, including both types at the transition volumes 4π/81 and 1/π.
OpenAI reports a complete higher-dimensional symplectic ball-packing criterion
- Mathematics
For n≥3, finitely many closed standard symplectic 2n-balls of positive capacities Rᵢ embed disjointly into the interior of a ball of capacity R exactly when the sum of Rᵢⁿ is below Rⁿ and every pair satisfies Rᵢ+Rⱼ<R.
OpenAI reports a positive solution to Tingley’s problem
- Mathematics
Every surjective isometry between the unit spheres of nonzero real Banach spaces extends uniquely to a surjective real-linear isometry of the full spaces.
OpenAI reports the Kirchberg–Rørdam character criterion
- Mathematics
For every nonzero unital separable complex C∗-algebra A and every free ultrafilter ω on the natural numbers, its norm central-sequence algebra has no characters if and only if A absorbs the Jiang–Su algebra under minimal tensor product.
OpenAI reports a single generator for separable-predual II₁ factors
- Mathematics
Every type II₁ factor with separable predual is generated as a von Neumann algebra by one bounded operator, equivalently two self-adjoint operators.
OpenAI reports vanishing of higher bounded Hochschild cohomology
- Mathematics
For every complex von Neumann algebra M and every integer k≥2, each bounded M-valued Hochschild k-cocycle has a bounded M-valued primitive, so Hᵏ_b(M,M)=0.
OpenAI reports universal strong Kadison–Kastler stability
- Mathematics
For every ε>0 there is a universal δ>0 such that any two unital von Neumann algebras on the same complex Hilbert space at Kadison–Kastler distance below δ are conjugate by a unitary u with norm distance from the identity below ε.
OpenAI reports quasi-isometric recognition of virtually polycyclic groups
- Mathematics
Every finitely generated group quasi-isometric to a finitely generated virtually polycyclic group is itself virtually polycyclic.
OpenAI reports a finitely presented simple amenable group
- Mathematics
There exists an infinite group that is finitely presented, simple and amenable.
OpenAI reports a dimension-free log-Sobolev inequality from subgaussian marginals
- Mathematics
Every centered log-concave probability measure with a Lebesgue density in R^n and linear a-subgaussian marginals satisfies a logarithmic Sobolev inequality with constant at most C a^2, uniformly in dimension.
OpenAI reports the planar Schrödinger convergence endpoint
- Mathematics
For initial data in H^(1/3)(R^2), the planar Schrödinger evolution converges almost everywhere to the data as positive time tends to zero, with the pointwise evolution defined by a Gaussian regularization limit first.
OpenAI reports critical three-dimensional wave local smoothing
- Mathematics
For the half-wave evolution in three spatial dimensions, the spacetime L3 norm on times [1,2] is bounded by the input L3 Sobolev norm with every positive derivative loss epsilon. The estimate yields the conjectured strict Euclidean local-smoothing range for all 2 < p < infinity.
OpenAI reports the bounded-data sphere restriction range
- Mathematics
The Fourier extension operator for the two-sphere maps bounded complex-valued data in L-infinity(S^2) to Lp(R^3) for every p > 3, with a constant depending only on p.
OpenAI reports the generalized Mukai conjecture
- Mathematics
Every positive-dimensional smooth complex Fano manifold of dimension n, Picard number rho, and pseudoindex iota satisfies rho(iota - 1) <= n; equality holds exactly for a product of rho copies of projective space of dimension iota - 1.
OpenAI reports the Global Spherical Shell conjecture
- Mathematics
Every connected minimal compact complex surface of class VII with positive second Betti number contains a global spherical shell.
OpenAI reports an embedded Zariski multiplicity counterexample
- Mathematics
Two reduced weighted-homogeneous complex hypersurface germs with isolated singularities are ambiently homeomorphic but have multiplicities two and three. They lie in one finite complex ambient dimension divisible by eight.
OpenAI reports canonical ampleness for compact hyperbolic Kähler manifolds
- Mathematics
Every positive-dimensional compact connected Kähler manifold with no nonconstant holomorphic map from the complex line has ample canonical bundle and is projective.
OpenAI reports a complex Lipman–Zariski counterexample
- Mathematics
There is a singular normal integral affine surface over the complex numbers whose tangent module is free of rank two on the whole surface.
OpenAI reports Bloch’s conjecture for complex surfaces with pg = 0
- Mathematics
For every smooth connected projective complex surface with geometric genus pg = 0, the Albanese map on integral degree-zero zero-cycles is an isomorphism. In the remaining general-type case pg = q = 0, this says the degree map CH0(S) to Z is an isomorphism.
OpenAI reports sharp nodal-length bounds on smooth surfaces
- Mathematics
For each fixed smooth closed connected Riemannian surface, the nodal set of every nonzero real Laplace eigenfunction of eigenvalue λ>0 has length at most C√λ, with C depending on the surface.
OpenAI reports Thomason model structures in all strict higher dimensions
- Mathematics
For every positive finite n and n=ω, small strict globular n-categories admit a proper combinatorial Thomason model structure Quillen equivalent to simplicial sets, with weak equivalences detected by the Street nerve.
OpenAI reports a ZFC counterexample to Naimark’s problem
- Mathematics
In ZFC there exists a unital infinite-dimensional simple complex C∗-algebra with a faithful tracial state whose nonzero irreducible representations are all unitarily equivalent, yet it is not an algebra of compact operators.
OpenAI reports a counterexample to Kaplansky’s quasitrace conjecture
- Mathematics
There is a separable unital complex C∗-algebra with normalized 2-quasitraces but no tracial state; its quasitraces fail additivity on a fixed pair of positive contractions.
OpenAI reports a QAC⁰ parity lower bound
- Theoretical Computer Science
For every fixed depth, polynomial qubit bound and ε>0, sufficiently large measured-output quantum circuits using arbitrary one-qubit and unbounded-arity Toffoli gates have some input on which parity succeeds with probability below 1/2+ε.
OpenAI reports the entropy photon-number inequality
- Mathematics
- Theoretical Physics
For two independent finite-mean-energy bosonic inputs with any finite positive number of modes, beam-splitter output satisfies N(ρC)≥ηN(ρA)+(1−η)N(ρB) for 0≤η≤1, where N(ρ)=g⁻¹(S(ρ)/n) is entropy photon number per mode.
OpenAI reports the Artin K(π,1) conjecture in finite rank
- Mathematics
The standard Salvetti complex of every Artin group with finitely many standard generators is aspherical, equivalently its universal cover is contractible.
OpenAI reports a nonresidually finite hyperbolic group
- Mathematics
There exists a torsion-free word-hyperbolic group that is not residually finite.
OpenAI reports Dixmier’s unitarizability equivalence
- Mathematics
A discrete group is amenable if and only if every uniformly bounded representation of it on a complex Hilbert space is similar to a unitary representation.
OpenAI reports a proof of the Boone–Higman conjecture
- Mathematics
A finitely generated group has decidable word problem if and only if it embeds in a finitely presented simple group.
OpenAI reports an Eilenberg–Ganea counterexample
- Mathematics
There is a finitely generated residually finite group of integral cohomological dimension two and geometric dimension three.
OpenAI reports single-fold Diophantine representations
- Mathematics
- Theoretical Computer Science
Every recursively enumerable subset of natural-number n-tuples, for n≥1, has an integer-polynomial Diophantine representation with exactly one complete natural-number auxiliary tuple for each member and none for nonmembers.
OpenAI reports nonuniqueness of percolation on nonamenable quasi-transitive graphs
- Mathematics
For every infinite connected locally finite nonamenable quasi-transitive graph under Bernoulli bond percolation, p_c<p_{2→2}≤p_u, yielding a nonempty parameter interval with infinitely many infinite clusters almost surely.
OpenAI reports the strong thin spanning-tree bound
- Mathematics
There is a universal C such that every finite loopless k-edge-connected multigraph on at least two vertices has a spanning tree meeting each nontrivial cut in at most C/k times the number of original cut edges.
OpenAI reports combinatorial invariance of Kazhdan–Lusztig polynomials
- Mathematics
Every order isomorphism between Bruhat intervals in arbitrary Coxeter systems preserves the corresponding equal-parameter Kazhdan–Lusztig polynomial.
OpenAI reports the Harary–Hill crossing-number formula
- Mathematics
For every n≥3, the ordinary crossing number of Kₙ equals ¼⌊n/2⌋⌊(n−1)/2⌋⌊(n−2)/2⌋⌊(n−3)/2⌋.
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arXiv · algebra-geometry — last successful metadata check: 21/10/2026, 07:20:46 UTC
arXiv · algorithms-formal — last successful metadata check: 21/10/2026, 07:19:48 UTC
arXiv · analysis — last successful metadata check: 21/10/2026, 07:21:51 UTC
arXiv · combinatorics — last successful metadata check: 21/10/2026, 07:18:49 UTC
arXiv · geometry-topology — last successful metadata check: 21/10/2026, 07:22:23 UTC
arXiv · logic-foundations — last successful metadata check: 21/10/2026, 07:23:04 UTC
arXiv · optimization-information — last successful metadata check: 21/10/2026, 07:23:10 UTC
arXiv · probability-dynamics-physics — last successful metadata check: 21/10/2026, 07:22:36 UTC
arXiv · theoretical-physics — last successful metadata check: 21/10/2026, 07:22:50 UTC