OpenAI reports superexponential van der Waerden lower bounds
- Mathematics
There is an absolute K such that W(r,k)>k^(k⌊log₂ r⌋/100000) for all k≥K and r≥2, so W(r,k)^(1/k) diverges for each fixed r≥2.
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1033 matching entries · Page 18 of 21
OpenAI reports a nonperiodic translational tile in dimension three
- Mathematics
A finite tile in ℤ³ admits a translational tiling but no fully periodic tiling; its unit-cube thickening has the same property in ℝ³ even with arbitrary real translations.
OpenAI reports positive area entropy for the standard map at large parameters
- Mathematics
The standard sine map of the two-dimensional torus has positive metric entropy with respect to normalized area for every sufficiently large positive parameter.
OpenAI reports that mixing implies all finite orders of mixing
- Mathematics
Every invertible mixing probability-preserving transformation is mixing of every finite order: for any k≥3 measurable sets, their translated intersection tends to the product of their measures as successive time gaps diverge.
OpenAI reports a superquadratic sensitivity separation
- Theoretical Computer Science
There are total Boolean functions with unbounded block sensitivity divided by sensitivity squared; in particular, a fixed exponent α>2 holds along a sequence with unbounded block sensitivity.
OpenAI reports sharp binary-information contraction on the discrete cube
- Theoretical Computer Science
For a binary channel on a uniform discrete cube under independent symmetric noise, a noisy coordinate maximizes retained information at fixed initial information; for Boolean functions this gives I(f(X);Y) ≤ 1 − h₂(ε) at crossover probability 0≤ε≤1/2.
OpenAI reports a depth-three circuit lower bound beyond every square-root constant
- Theoretical Computer Science
One fixed polynomial-time Boolean language has exact depth-three OR–AND–OR circuit size greater than 2^(A√n) for every fixed A>0 at all sufficiently large input lengths n.
OpenAI reports a one-sample matroid prophet inequality against full-seed ordering
- Theoretical Computer Science
For every finite matroid, one independent sample per element supports an online rule with expected reward at least 2^-310 times the expected offline optimum, even when the arrival-order adversary sees all samples, online values, and the rule’s seed.
OpenAI reports a doubling Hilbert set with no finite-dimensional bi-Lipschitz embedding
- Mathematics
A subset of real Hilbert space has doubling constant at most 76,800 yet admits no bi-Lipschitz embedding into any finite-dimensional Euclidean space at any finite distortion. Moreover, every infinite-dimensional real Banach space contains a compact doubling subset with the analogous obstruction and a universal doubling bound.
OpenAI reports the Euclidean Steinitz–Bergström bound
- Mathematics
Every prescribed-order finite sequence of vectors in the Euclidean unit ball of R^d has one choice of signs whose every signed prefix has norm at most C sqrt(d), with C independent of sequence length. Consequently every indexed zero-sum family admits an ordering with every unsigned prefix bounded by C sqrt(d).
OpenAI reports subpolynomial dimension reduction in Lp
- Mathematics
For each fixed real 1 < p < infinity with p != 2 and distortion D > 1, every n-point subset of a real Lp space has a nonlinear embedding into finite-dimensional l_p with dimension at most exp(C(p,D) (log n)^gamma), where gamma = 2-p for p < 2 and 1-2/p for p > 2.
OpenAI reports the logarithmic Brunn–Minkowski inequality
- Mathematics
For all origin-symmetric convex bodies K and L in every dimension and every interpolation parameter t in [0,1], the volume of their logarithmic Wulff combination is at least |K|^(1−t)|L|^t.
OpenAI reports universal energy optimality of the triangular lattice
- Mathematics
The density-one triangular lattice minimizes lower energy per particle among all locally finite planar configurations of centered-disk density one for every nonnegative completely monotone function of squared distance, including infinite-energy comparisons.
OpenAI reports Petty’s projection-volume inequality in dimensions at least four
- Mathematics
For every convex body K in Rⁿ with n≥4, its normalized projection-body volume is at least the ellipsoid value, with equality exactly for ellipsoids.
OpenAI reports an endpoint gradient bound for centered disk averages
- Mathematics
For every real f in W¹,¹(R²), the centered-disk maximal function has a globally integrable weak gradient and satisfies ||∇Mf||₁ ≤ C||∇f||₁ for an absolute constant C.
OpenAI reports uniform Hilbert-transform bounds for Lipschitz directions
- Mathematics
For a planar Lipschitz unit vector field with Lipschitz constant K>0, short Hilbert transforms with outer scale at most 1/(10⁶K) have a universal L² bound independent of the inner hard truncation, including fields depending on both coordinates.
OpenAI reports a higher-codimension Riesz-transform criterion for rectifiability
- Mathematics
For d≥4 and 2≤n≤d−2, an n-Ahlfors–David regular measure in Rᵈ with uniformly L²-bounded positive hard truncations of its n-dimensional Riesz transform is uniformly n-rectifiable.
OpenAI reports almost-everywhere Fourier convergence in L log L
- Mathematics
For every complex-valued function on the circle with finite integral of |f| log(2+|f|), its ordinary symmetric Fourier partial sums converge almost everywhere to f along the full sequence.
OpenAI reports compatible universal-cover splitting for Kähler manifolds
- Mathematics
If the tangent bundle of a compact connected Kähler manifold splits into two positive-rank integrable holomorphic subbundles, its ordinary universal cover is a product whose tangent factors lift the specified summands.
OpenAI reports ample rank-two bundles without Griffiths-positive metrics
- Mathematics
On P¹ × P¹, coordinatewise power pullbacks of an explicit rank-two bundle, tensored with O(1,1), are ample for every positive power yet admit no smooth strictly Griffiths-positive Hermitian metric for all sufficiently large powers.
OpenAI reports a four-variable stable-coordinate counterexample
- Mathematics
An explicit degree-five polynomial in four complex variables is not a coordinate there but becomes a coordinate after one new variable; each fiber is affine three-space without an ambient rectification.
OpenAI reports a complex affine-space cancellation counterexample
- Mathematics
There is an explicit integral complex affine fourfold X such that X × A¹ is isomorphic to A⁵ but X is not isomorphic to A⁴.
OpenAI reports prime-degree torus-packet equidistribution
- Mathematics
For each fixed prime degree at least five, volume-weighted packets of totally real fields equidistribute to Haar measure as their multiplier-order discriminants grow, with no escape of mass, arbitrary local homothety type, and varying fields allowed.
OpenAI reports polynomial mixing of graph switches for every degree sequence
- Theoretical Computer Science
For every graphical degree sequence on n≥4 vertices, the lazy simple-undirected switch chain that proposes switches uniformly on four vertices has total-variation mixing time at distance 1/4 at most 2n⁸.
OpenAI reports polynomial-time optimal three-machine unit-job scheduling
- Theoretical Computer Science
A deterministic polynomial-time algorithm finds a minimum-makespan schedule for unit-length jobs with arbitrary acyclic precedence constraints on three identical parallel machines.
OpenAI reports an FPRAS for counting perfect matchings in general graphs
- Theoretical Computer Science
A randomized algorithm approximates the number of perfect matchings in every finite simple undirected graph to relative error ε with failure probability at most δ in worst-case bit time polynomial in the input length, ε⁻¹, and log δ⁻¹; it returns zero with certainty when there is no perfect matching.
OpenAI reports a cubic permanent–determinant lower bound
- Theoretical Computer Science
The complex border determinantal complexity of the m×m permanent is Ω(m³), allowing affine-linear determinant representations and coefficientwise limits.
OpenAI reports hardness of large independent sets in three-colorable graphs
- Theoretical Computer Science
For every fixed 0<δ<1/3, it is NP-hard to distinguish three-colorable finite simple graphs from graphs whose largest independent set has fewer than δ times the vertices.
OpenAI reports perfect completeness for 2-to-1 games
- Theoretical Computer Science
For every fixed rational 0<δ<1, it is NP-hard to distinguish satisfiable 2-to-1 games from unweighted games of value at most δ, with an alphabet depending only on δ.
OpenAI reports projective-space bases for Lagrangian fibrations
- Mathematics
The normal projective base of any projective Lagrangian fibration from a compact irreducible holomorphic symplectic Kähler manifold is projective space.
OpenAI reports the ordinary-double-point volume gap
- Mathematics
A singular n-dimensional complex algebraic klt germ with zero boundary has normalized volume at most 2(n−1)^n, with equality exactly at an analytic ordinary double point.
OpenAI reports positive lower density of large prime gaps
- Mathematics
For every fixed C>0, a positive lower proportion of consecutive prime gaps pₙ₊₁−pₙ exceed C log pₙ.
OpenAI reports regular variation of the number of totient values
- Mathematics
If V(x) counts distinct values of Euler’s totient function up to x, then V(cx)/V(x) tends to c for every fixed real c>0.
OpenAI reports an unconditional first moment for cubic Gauss sums
- Mathematics
Over all primary Eisenstein primes of norm at most X, the sum of normalized cubic Gauss sums has the sharp-cutoff asymptotic (6/5)c₍*₎X^(5/6)/log X, where c₍*₎=(2π)^(2/3)/(3Γ(2/3)).
OpenAI reports a squarefree-value density for every irreducible quartic
- Mathematics
Every irreducible integer quartic with no fixed prime-square divisor takes squarefree values with the predicted positive Euler-product density.
OpenAI reports the Margulis–Platonov conjecture over global function fields
- Mathematics
For an absolutely almost simple simply connected algebraic group over any global function field, including characteristic two, every noncentral abstract normal subgroup of its rational points is the inverse image of an open normal subgroup of the finite product of its anisotropic local groups.
OpenAI reports the abelian Zilber–Pink conjecture
- Mathematics
For every irreducible subvariety X of an abelian variety over a number field, there are only finitely many maximal atypical subvarieties of X relative to its smallest containing torsion coset.
OpenAI reports independent largest prime factors of consecutive integers
- Mathematics
For every fixed 0<a,b<1, the natural density of integers n with P⁺(n)≤nᵃ and P⁺(n+1)≤nᵇ is ρ(1/a)ρ(1/b), where P⁺ is the largest prime factor and ρ is the Dickman function.
OpenAI reports a 21 move Maker strategy for Snaky
- Mathematics
In the initially empty infinite square-board weak Maker–Breaker game, Maker can complete a translated, rotated or reflected Snaky hexomino within 21 legal Maker claims against every legal Breaker play.
OpenAI reports a linear cycle-and-edge decomposition for every graph
- Mathematics
- Theoretical Computer Science
For some absolute constant C, every finite simple undirected graph on n vertices has an edge partition into at most Cn simple cycles and single edges.
OpenAI reports a proof of Barnette’s conjecture
- Mathematics
- Theoretical Computer Science
OpenAI reports that every finite simple cubic bipartite planar graph that is 3-vertex-connected has a Hamiltonian cycle.
Human–ChatGPT collaboration reports harmonic norms over arbitrary complete non-Archimedean fields
- Mathematics
Oren Ben-Bassat reports that every polystable representation of a finite quiver over any complete non-Archimedean field admits a split HKKP harmonic norm, without spherical-completeness or value-group restrictions.
AI-assisted certificate reports optimality of logarithmic-spiral shoreline search
- Mathematics
- Theoretical Computer Science
Alexander Temerev reports that every planar search path for an unknown straight shoreline has competitive ratio at least 13.8111351794611…, matching the logarithmic spiral. The proof permits paths with decreasing radius or reversing polar angle.
Kourovka Problem 21.150: Counterexample to a rank bound for p-group extensions
- Mathematics
A 3-group extension G=A⋊B with elementary abelian A and B and trivial C_B(a) can have rank(Z(H)∩H′)>rank(B), where H=⟨a,B⟩.
Kourovka Problem 21.147: Non-abelian counterexample for relatively convex subgroups
- Mathematics
In a non-abelian right-orderable group, the right-relatively convex subgroups need not form a sublattice of its subgroup lattice.
Kourovka Problem 21.24: Finite-group cograph power graphs are chordal
- Mathematics
Every power graph of a finite group that is a cograph is chordal; equivalently, an induced four-cycle forces an induced four-vertex path.
Kourovka Problem 21.8: Class transpositions generate the full symmetric group
- Mathematics
For k ≥ 4, horizontal class transpositions of moduli at most k generate a group isomorphic to the symmetric group on lcm(2,…,k) letters.
Kourovka Problem 20.125: A surjective non-injective Rota–Baxter operator on a non-abelian group
- Mathematics
A non-abelian group admits a weight-one Rota–Baxter operator that is surjective but not injective, answering Problem 20.125.
Kourovka Problem 19.25: Order and totient sum do not determine simplicity
- Mathematics
A simple group and a non-simple group both have order 6048 and totient sum 23984, disproving the proposed simplicity test in Problem 19.25.
Kourovka Problem 18.50: Every permitted count of permuted products
- Mathematics
For every positive n and 1 ≤ k ≤ n!, a group has n distinct elements whose n! ordered products take exactly k distinct values.
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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