Convex bodies, distances, packings, coverings and discrete geometric configurations.
In every arbitrary two-coloring of the Euclidean plane, each non-equilateral three-point set, including three distinct collinear points, has a monochromatic congruent copy. At most one equilateral side length can be missed, and an alternating-strip coloring shows that one exception can occur.
An OpenAI manuscript claims a real homogeneous hyperbolic polynomial whose closed cone has no exact finite affine semidefinite lift, even with arbitrary real coefficients and any finite number of auxiliary variables.
For every simple graph G on seven vertices, either G or its complement has an injective unit-distance representation in Euclidean R³, where every edge has length one and nonedges are unconstrained. This proves the d=3 case of Frankl–Kupavskii–Swanepoel Problem 2; their general 2d+1-vertex question remains open. Combining with the published lower bound gives f_D(7)=3, although the lower bound is outside the formal target.
For every closed simple polygon with n vertices, including collinear vertices and equal heights, an explicit exact-real-RAM program returns a conforming triangulation with n−2 triangles in at most C(n+1) steps and natural words bounded by K(n+1). It is a variant of Chazelle’s algorithm that completes the visibility map directly below n=2³⁰+2.
For every nonempty cograph G, its Hansen polytope in dimension d=|V(G)|+1 has normalized Mahler volume at least one, at least 3^d nonempty faces and at least 2^d d! full flags. Each equality holds exactly for threshold graphs.
For a well-formed Mazur edge, cycle, series and join scheme with n≥2 vertices and j nontrivial joins, the span satisfies T+1≤2^j(n−2j), with equality at (n,j)=(2,0) or 2j+3≤n. Its worst case is T=2^(n/2)−1 for even n and T=3·2^((n−3)/2)−1 for odd n. Companion theorems give interior dart mass ≤aₛbₛT, Dirichlet row norm ≤2(n−1)²T and |det M|≤(2(n−1)²T)^n, bound the candidate products T²|det M| and T³|det M|, and prove z=adj(M)u with Mz=Δu and boundary zₛ=Δhₛ(T−hₛ) when hₛ≤T, where Δ=det M.
Strinz reports that every finite set of distinct points on a circle is Euclidean Ramsey when all but at most three points have coordinates algebraic over Q. The circle’s center and radius may be arbitrary real numbers.
The AI-origin improvement in Appendix B gives non-expansive line embeddings with O(log h) average distortion for the stated vertex-weighted metrics of connected graphs excluding the one-subdivision of K_h as an induced minor. The paper derives balanced separators of size O(h log h √m) for the corresponding m-edge graphs.
Möbius invariance and the stated corollaries for finite distinct spherical Ramsey configurations, relative to the credited OpenAI classification and classical identification of the Möbius action.
For every s≥3, allowed base q and k≥1, the explicit N=q^(sk) digital net has covering radius ≤√s N^(−1/s); when q≥5a*_s+4 its point separation is ≥(1−a*_s/q)N^(−1/s), with the stated bound on a*_s.
Every compact connected planar set that can be moved continuously through both a right-turning and left-turning unit-width hallway from the same starting orientation has area at most M=1+4Y²+arctan(Y), where Y>0 solves 4Y³+3Y=1. Equality holds exactly for rotated and translated copies of Romik’s sofa, and the sofa attains M.
Vu-Le reports that every compact connected planar shape movable through both left- and right-turning unit corridors from the same initial orientation has area at most 1.64495…; equality identifies Romik’s sofa up to rigid motion.
The-Anh Vu-Le’s project reports that Gerver’s sofa is the unique area-maximizing moving sofa up to rigid motion, and that sofas with area deficit ε are within C√ε after translation in its Hausdorff-distance model, with exponent 1/2 optimal.
Every set of six distinct points on a circle satisfies the FieldCriterion tensor condition of the OpenAI Euclidean Ramsey classification. More generally, every spherical finite set of at most m+4 points, where m is its affine-span dimension, satisfies that criterion in its own coordinate field; every spherical set with algebraic coordinates also satisfies it.
An explicit homogeneous hyperbolic polynomial of degree 20 in 23 real variables has a closed hyperbolicity cone that cannot be represented as the positive-semidefinite region of any finite real symmetric linear matrix pencil in the original variables.
Every proper unit-distance colouring of the plane whose unit ball is the regular 14-gon uses at least five colours.
For every n≥2 and real K, each regular point of a noncollapsed RCD(K,n) space has an ambient-open neighborhood bi-Lipschitz to an open subset of Euclidean n-space, with distortion bounded only by n.
For every fixed dimension d≥3, a constant c_d>0 makes every n-point set of distinct points in ℝᵈ determine at least c_d n^(2/d) distinct distances.
In each dimension, every convex body has isotropic constant no greater than that of a simplex, with equality only for simplices.
A finite Euclidean configuration is Ramsey for monochromatic congruent copies at its original scale exactly when its coordinate field admits the paper’s rational-tensor matrix satisfying the pointwise evaluation identities and identity-matrix product identities; the criterion covers every affine dimension.
For every fixed ε>0, all but o(n) points of any n-point planar set determine at least n^(1−ε) distinct nonzero distances as n grows, uniformly over the set.
There are separable real Banach spaces X and Y with a global bi-Lipschitz bijection but no bounded linear isomorphism between them; X contains an isometric copy of c₀(ℓ₂), while Y contains no bounded linear copy of that space.
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).
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.
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.
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).
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.
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.
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.
For every convex body K in R^n, n≥1, the manuscript reports volume product at its Santaló point at least (n+1)^(n+1)/(n!)², with equality exactly for simplices.
OpenAI reports that every origin-symmetric convex body in n-dimensional real space has volume product at least 4^n/n!, with equality precisely for invertible linear images of Hanner polytopes.
Bizeul, Klartag and Lehec report a universal upper bound for the Poincaré constant of every isotropic log-concave probability measure in every dimension, resolving the KLS conjecture in its functional formulation.
For five, six and seven freely rotated, interior-disjoint unit squares in a closed disk, the project reports exact least radii √(5/2), √q* and √13/2, respectively, where q* is its specified quartic root. It classifies all optimal packings up to rotation and relabelling: the five- and six-square optima are unique, while seven squares admit a three-parameter family.
The model-origin optimality arguments for five through seven freely rotated unit squares in a disk, separating known small cases and later human commentary.
For four positive masses in the planar Newtonian problem, each cyclic order has exactly one strictly convex central configuration up to similarity. The v2 theorem also gives Q≥K/4 at every convex configuration, nondegenerate local minima and analytic dependence.
The manuscript reports matching bounds exp(Θ(√(log d · log log d))) for the least ℓ₁ distortion of unit-cost edit distance on strings of length at most d.
A separable real Banach space Z has no closed linear copy of c₀ yet is globally bi-Lipschitz equivalent to Z×c₀. The construction also yields bi-Lipschitz universality for separable metric spaces.
Every subset of R⁴ containing a unit segment in every direction has Hausdorff dimension four, without a measurability, compactness or stickiness condition.
For the manuscript’s unit-background planar Coulomb renormalized-energy class and admissible square-cutoff families, every admissible field has energy at least that of the density-one triangular lattice. A prior equivalence then identifies the linear term of optimal spherical logarithmic energy.
For every sufficiently large dimension n, an OpenAI manuscript constructs a centrally symmetric convex body whose unrestricted translative covering density exceeds c n log n for an absolute c>0. With Rogers’ O(n log n) upper bound, this fixes the worst-case unrestricted order and rules out a universal O(n) bound.
There are constants C>0 and 1≤β<4/3 such that every finite n-point planar set has at most Cn^β unordered unit-distance pairs.
A real radial Schwartz function on the plane has nonnegative Fourier transform, is nonpositive beyond unit distance, and has normalized origin values making the Cohn–Elkies linear-programming bound attain the triangular-packing density π/(2√3).
An Artin group on 116 generators, with off-diagonal labels only 2, 3 or infinity, has no proper cocompact isometric action on any nonempty proper CAT(0) metric space, in any dimension.
The manuscript reports an L₁ embedding with a universal distortion bound for every finite connected planar graph with positive real edge lengths.
The manuscript reports density at most C n log n for a single-lattice covering by translates of every convex body in dimension n ≥ 2.
The manuscript reports a nine-dimensional counterexample using rank-one orthogonal projectors on R⁴ with the Frobenius metric.
The manuscript reports that five colors cannot suffice, implying that the plane’s chromatic number is six or seven if the result holds.
For every origin-symmetric compact convex body K with nonempty interior in Rⁿ, n≥2, the Gromov width of int(K)×int(K°) is 4. Every symplectic ball of capacity c<4 embeds.
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.
For every transcendental radius r>2, a seven-point set on a circle of radius r, including r=π, has a finite coloring of Euclidean space with no monochromatic congruent copy in any dimension.
Tsiokos reports a simply connected rational polyhedral tile that tiles R³ by congruent copies, reflections allowed, while every tiling has no infinite-order symmetry. Each tiling has a unique nested supertile hierarchy and a finite symmetry group of order at most 24.
For every real p>4, Xiong constructs an equilateral set of 8m points in ℓₚ^(8m−2) for some power-of-two m, exceeding the proposed dimension-plus-one maximum.
For every positive dimension n, the Euclidean kissing number τₙ is an attained maximum and satisfies τₙ≤(√6)ⁿ. A planar hexagon gives τ₂=6, so √6 is the least base in a bound τₙ≤αⁿ valid in every dimension without a prefactor.
For every real Hilbert space H, including nonseparable spaces, the one-dimensional Besicovitch rectifiability threshold σ₁(H) is at most 0.6934. In particular, a planar Borel set of finite H¹ measure is countably one-rectifiable when its lower one-density exceeds any fixed threshold greater than 0.6934 almost everywhere.
For every sufficiently large genus g, Cai constructs a closed hyperbolic surface with systole at least log g−12 log log g. The lower limit of the largest possible systole divided by log g is therefore at least one.
The three exact model-generated 604-point kissing configurations in dimension 11 in Station v2 Section 6.3. Two are reported as new isometry classes; the overlapping concurrent construction does not establish sole AI priority.
Every finite planar point set of at least 10^11055931 points has four collinear points or six pairwise visible points.
In each of dimensions four and five, for every sufficiently small nonzero ε, an explicit smooth odd function on S^(d−1) cannot agree with a linear functional on any rotated normalized A_d root system. Through Kuperberg’s equivalence, these give counterexamples to Makeev’s universal-cover conjecture in those dimensions.
For odd n≥3, Lu and Yang obtain a global Lipschitz family of exact hyperplane-section isometries after a positive-degree bundle pullback. Combined with their degree argument, this proves that a real Banach space whose n-dimensional subspaces are all linearly isometric is Hilbert when 2≤n<dim X.
There are noncongruent simply connected strictly convex Euclidean plane domains with real-analytic boundaries and identical Steklov spectra, including multiplicities. Such pairs can approach a disk smoothly.
For n independent standard Gaussian points in R^d, a positive semidefinite S with x_iᵀSx_i=d for every i exists with high probability when limsup n/d²<1/4 and fails to exist with high probability when liminf n/d²>1/4.
There are full-dimensional 0/1-polytopes of dimension 4n+1 whose graph edge expansion decreases exponentially with dimension.
Liu claims that a full-dimensional compact convex body in Rⁿ with barycenter zero, zero as its only interior lattice point and volume (n+1)ⁿ/n! is an integral unimodular image of the centered simplex (n+1)Δₙ−(1,…,1).
For every relative binary distance 0<δ<1/2, Chapter 2 gives an asymptotic rate bound strictly below the classical MRRW bound. Companion spherical-code bounds improve the localized linear-programming hierarchy for 0<s<1.
A direct 3SAT reduction proves NP-hardness of approximating Euclidean closest vector within n^(1/400), under the dimension and promise conventions of Chapter 7. The chapter gives separate consequences for binary nearest codeword and finite p norms.
For the optimal Cohn–Elkies linear-programming packing bound LP_d, OpenAI gives the exponential root limit √(e/(2π)). This yields the sphere-packing upper bound (√(e/(2π))+o(1))ᵈ and corresponding asymptotic Fourier sign-uncertainty radii.
For every positive dimension n, a compact convex body with nonempty interior, barycenter 0 and no interior lattice point other than 0 has volume at most (n+1)ⁿ/n!. A centered simplex attains equality. This is the inequality, separate from the later equality classification.
The Schubitope in R¹⁴ from columns {8,…,14} and {12,13,14}, equivalently Δ(7,14)+Δ(3,14), has Ehrhart quadratic coefficient −687323/79200. It is therefore not Ehrhart positive.
For every α>0 and three positive point charges in R^n with n≥2, the potential has at most six nondegenerate equilibria. The new step proves a four-contact statement using separation at the unique saddle of a first integral.
Five positive point charges in R³ can have at least 24 nondegenerate electrostatic equilibria. A small perturbation makes all critical points nondegenerate and finite in number, exceeding Maxwell’s proposed (n−1)² bound of 16 when n=5.
For any planar convex body K and 0<W<width(K), one plank of width W covers at least as much area of K as any finite family of planks whose total width is at most W. The assessed AI-associated result is this planar reduction.
Domina and Ceriotti give noncongruent atom-centered environments with identical complete correlations through seven-neighbor order, including three-dimensional lifts. For every finite correlation order and angular cutoff, they also construct indistinguishable noncongruent environments.
Temerev and Doria report that the minimum escape-path length for the golden gnomon, an isosceles triangle with unit equal sides and 36-degree base angles, is 1.282676025459… . A symmetric seven-piece path attains the value, and their lower bound covers every rectifiable competitor.
For every isotropic log-concave random vector X and symmetric matrix M, Var⟨MX,X⟩ ≤ 2 E|∇⟨MX,X⟩|². Letwin also derives a general KLS bound of order (log n)^(1/4).
Pohoata constructs n incident point-line pairs in [0,1]² whose off-diagonal distances are at least n^(−2/3−ε) for every ε>0 and all sufficiently large n. With the previous upper bound, the maximum separation is n^(−2/3+o(1)).
Star Fleet's construction shows that for every odd n at least 201, every integer k at least n·2^n·ceil(49n/100)^n+2 is the count of a tiling of the unit n-cube by positive homothetic cubes. This gives c(n)=o(n^n) along odd dimensions and refutes the proposed universal lower bound c(n)≫n^n; it does not determine the best bounds for c(n) in all dimensions.
Jumagulov certifies that the dimension-36 Cohn–Elkies two-point LP optimum is at least 146.1036734821, more than 32.91 times the Kschischang–Pasupathy packing’s center density.
Gargava gives an exact rotated E8 lattice packing of B₂⁴ × B₂⁴ with density π⁴(1+1/√5)⁴/1024, about 9.67% above the product D4 packing. A separate determinant-preserving perturbation of the nine-dimensional densest lattice Λ9 has lower computed flat-torus height.
Five simple 3-polytopes with 10 faces violate the printed IRIS Conjecture 6.1. One has four triangular, three pentagonal and three heptagonal faces, but no hexagonal faces, below the conjectured lower bound of 1/5.
In Korsky’s upper-bound argument for north-east lattice paths with no k collinear visited vertices, the mediant of Farey neighbors raises the uniform density-increment product from 1/[8(k−1)²] to 1/[4(k−1)²]. It gives log A(k)≤(2/e+o(1))(k−1)².
Fourteen binary vertices form a simplicial seven-dimensional 0/1-polytope with the maximum 2d vertices but without central symmetry, countering the proposed cross-polytope characterization.
Finite planar lattice sets can have an arbitrarily large ratio of ordered rectangles to ordered isosceles triangles, including degenerate configurations. Bennett, Kovač, Nakamura and Oliveira quantify the construction in a box [-N,N]² by a ratio at least (log N)^(c/log log log N), with an analogous cardinality bound.
An explicit reduced planar convex body of thickness 1, with support-function parameters m=15/16 and ε=13/20, has area 0.7862156027…>π/4. It refutes Lassak’s proposed bound area(R)≤(π/4)Δ(R)² for all reduced planar bodies.
Under OVH or SETH, Saha, Xu and Ye prove n^(2−o(1)) lower bounds for exact Furthest Pair, Bichromatic Closest Pair, Maximum Inner Product and Hopcroft’s problem in every efficiently constructible superconstant dimension.
Ajwani, Gajjala, Raman and Ray give 64 axis-parallel rectangles with packing number 16 and piercing number at least 32, disproving Wegner’s proposed bound 2ν−1. Their package construction yields piercing and clique-LP ratios approaching 5/2.
AlphaEvolve found 51- and 151-unit-disk configurations whose odd-covered areas are estimated by Monte Carlo calculations at about 2.73 and 2.51, respectively.
Korsky proves that every sufficiently large prescribed length L admits full-chord sets in each compact convex body with finite piecewise C² boundary and discrepancy O((log L)^(3/2)). For the disk, every full-chord set has discrepancy Ω(log L).
The human-verified writeup gives a fixed ε>0 and growing planar point sets with at least n^(1+ε) unit distances.
For n=4 and p=2, the AlphaEvolve-found cosine correlation matrix Σᵢⱼ=cos(π(i−j)/n) has a strictly smaller expected squared minimum absolute Gaussian coordinate than the regular-simplex correlation matrix.
Portier proves that an edge-minimal generically globally rigid graph in Rᵈ with at least d+3 vertices contains no K_(d+2) subgraph. Equivalently, the only minimally globally rigid graph containing that clique is K_(d+2) itself.
For the largest measure M(R) of a measurable subset of a radius-R planar disk with no positive integer distance between distinct points, M(R)≤C√R. Together with Sárközy’s earlier lower construction, this yields M(R)=R^(1/2+o(1)).
For infinitely many n=q+1 with q a prime power, Ho constructs n points in R^(n²−n) whose distances for distinct pairs differ by at least one but whose diameter is at most (1−1/π²+o(1))n².
For integers r≥3, k≥4 and n≥72, the maximum ordinary-line count Fᵣ,ₖ(n) is at least n²/12−(10/3)n. The elliptic-curve construction has no four collinear points and a bipartite ordinary-line graph.
A 1,280-word binary code of length 19 and minimum distance five inside a five-punctured extended Golay code extends Cohn and Li’s 10,668-point configuration. It gives the lower bound k(19)≥11,948.
A centrally symmetric simplicial six-dimensional polytope with 48 vertices has an antipodal boundary quotient that triangulates real projective five-space on 24 vertices.
Aletheia constructed planar n-point sets with no four cocircular in which every point determines fewer than 3n/4 distinct distances. These refute the near-n-distances question under that hypothesis.
Harmonic reports that Aristotle generated machine-verified solutions for 2025 IMO Problems 1–5 from human-formalized problem statements, with the Lean proofs published.
An advanced Gemini Deep Think model produced natural-language solutions to five of the six 2025 International Mathematical Olympiad problems, earning 35 of 42 points under the official coordinators’ grading.
AlphaEvolve’s authors report an eleven-dimensional kissing configuration with 593 outer spheres.
AlphaGeometry solved 25/30 specialized IMO-AG-30 classical Euclidean geometry translations using learned auxiliary-point constructions plus a symbolic deduction and algebra engine; the paper does not claim all natural-language IMO geometry conditions are represented.
Convex bodies, distances, packings, coverings and discrete geometric configurations.
Publication dates and editorial changes.
Revisiting Separators in String Graphs, Region Intersection Graphs, and Induced-Minor-Free Graphs
Every two-coloring of the plane misses at most one triangle, which is equilateral
OpenAI reports bi-Lipschitz c₀ absorption without a linear c₀ copy
OpenAI reports a hyperbolicity cone with no finite semidefinite lift
AI-assisted formal proof reports four-body convex uniqueness
OpenAI claims triangular minimality for planar Coulomb energy
Astra proof gives polynomial-factor hardness for closest vector
AI-built Lean program triangulates simple polygons in linear time
OpenAI reports a superlinear translative covering-density lower bound
Danus-assisted proof characterizes equality in Ehrhart’s volume bound
ChatGPT-found proof excludes large cliques in minimally rigid graphs
Claude helps reduce the three-charge equilibrium bound to six
GPT-suggested five-charge construction refutes Maxwell bound
GPT-origin construction gives the minimal-distance sharp exponent
ChatGPT-found constructions challenge two geometry-of-numbers claims
Star Fleet reports a counterexample to Erdős Problem 769’s lower bound
Bakaev and Polyanskii report one-plank optimality in the plane
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