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639 source-grounded records. Verification labels remain separate from source authentication and publication status.
Imported from VibeMathed under CC BY 4.0. Each record links to its registry entry and named primary source. Registry verification is preserved verbatim.
A Cayley graph is minimal when no proper subset of its connection set generates the group. Babai asked whether minimal Cayley graphs have bounded chromatic number. Resolved negatively: finite minimal Cayley graphs exist with arbitrarily large chromatic number.
proved in dimension at most three
the construction is probabilistic; an explicit uniformly samplable example remains open
Refuting log-concavity of the flat counts is weaker than refuting their unimodality, since log-concavity is the stronger property. A counterexample to unimodality followed three weeks later and is tracked separately as Rota's Unimodality Conjecture for Matroid Flats; this paper came first.
Constructed an explicit class of smooth random, time-dependent incompressible velocity fields on T^3, obtained by alternating smooth shear flows with iid random phases on finite time blocks. For every fixed sufficiently small resistivity, the magnetic field has an almost-sure exponential growth rate at least 1/2, together with a time-uniform lower bound whose random prefactor has a resistivity-uniform inverse-mome…
The swap chain flips checkerboard 2×2 blocks to sample 0/1 matrices with fixed row and column sums. Kannan, Tetali and Vempala conjectured in 1997 that it mixes in polynomial time for all feasible margins; the lazy chain is shown to have spectral gap at least $\binom{m}{2}^{-1}\binom{n}{2}^{-1}$ on $m \times n$ matrices, which is worst-case tight and settles the bipartite case.
The Benjamini-Hochberg procedure is known not to control the false discovery rate at its nominal level under arbitrary dependence. A folklore conjecture in the FDR literature held that it must at least control the FDR up to a universal multiplicative constant. It does not: there are finite Gaussian models whose FDR divided by $q$ diverges as $q \downarrow 0$, with an explicit two-sided lower bound $q\sqrt{\log(1/q…
For fixed $\delta\in(0,1)$ and all sufficiently large $\Delta$ depending only on $\delta$, Glauber dynamics for proper $q$-colorings mixes rapidly on every graph of girth at least $5$ whenever $q\ge(1+\delta)\Delta$: spectral gap $\Omega_\delta(1/n)$ and $t_{\mathrm{mix}}(\varepsilon)=O_\delta(n^2\log q+n\log(1/\varepsilon))$. An analogous theorem holds for the anti-ferromagnetic Potts model at $q\ge(1+\delta)(1-\…
a structured special case, proved the same week as the independent group version
Closes both gaps left open by Bandeira and Maillard: exact fitting, and removal of the operator-norm constraint. The threshold turns out to be governed by the statistical dimension d(d+1)/4 of the PSD cone.
The big-line-big-clique conjecture of Kára, Pór and Wood asserts that for all $k, \ell$ there is an $n$ such that every finite point set of at least $n$ points contains $\ell$ collinear points or $k$ points that pairwise see each other. True for $\ell = 4$, $k = 6$, the first case left open: every finite point set of size at least $10^{11055931}$ has four collinear points or six pairwise visible points.
What is the best deterministic polynomial-time approximation ratio for the permanent of a Hermitian positive semidefinite matrix? Resolved up to lower-order terms in the exponent: an explicit concave maximisation $\widehat P(A)$ satisfies $e^{-\gamma n}\widehat P(A) \le \mathrm{per}(A) \le \widehat P(A)$, giving a deterministic $e^{(\gamma+\varepsilon)n}$-approximation for every $\varepsilon > 0$ and matching the…
Koivisto asked at Dagstuhl in 2013 whether the linear extensions of an arbitrary $n$-element poset can be counted exactly in time $O^*(c^n)$ for some $c < 2$. Yes: a deterministic exact algorithm runs in $O^*(1.89^n)$, breaking the $2^n$ barrier for the general problem.
For the Ising pure $p$-spin glass where $p \geq 3$, it was predicted by Gardner that there exists critical inverse temperatures $0<\beta_1^p<\beta_2^p <\infty$ such that: (1) When $0<\beta\leq \beta_1^p$, the Parisi measure is replica symmetric (RS); (2) When $\beta_1^p<\beta \leq \beta_2^p$, the Parisi measure is one-step replica symmetry breaking (1-RSB); (3) When $\beta>\beta_p^2$, the Parisi measure is full r…
If a bilinear form admits an $(r,s)$-sparse bound, its coordinate-wise extension to $\mathbb C^n$-valued functions admits an $(r,s)$-convex body sparse bound, for $1\le r,s<\infty$ with $\tfrac1r+\tfrac1s>1$. It holds both in a fixed dyadic lattice (Theorem 2.6, constants independent of the ambient dimension) and for arbitrary cubes (Theorem 3.4), via a randomization of the good part of the form. On scope, two th…
At the conjectured density, must every $k$-uniform hypergraph contain a short nontrivial even cover - a set of hyperedges covering each vertex an even number of times - with no superfluous polylogarithmic factors? Known up to polylog factors since 2022; now proved exactly for every $k \ge 3$.
rules out the two-point LP method in this dimension; the optimal packing in dimension 36 remains unknown
in the range s in (1/4,1); Brezis's Problem 5.4 outside that range is untouched
Give an explicit profinite presentation of $\operatorname{Gal}(\overline{\mathbb{Q}}_2 / \mathbb{Q}_2)$. The tame local cases were settled by the early 1980s; the dyadic case was the last one missing. The new presentation has four generators, two word relations and a pro-$2$ condition on the wild generators.
The paper constructs an exact cluster $F\subseteq\mathbb{Z}^2$ of cardinality 8 with full affine span and an $F$-tiling whose orbit closure contains no 1-periodic $F$-tiling, giving a non-degenerate counterexample to Nivat's conjecture for non-convex windows. This answers negatively a question of Kari and Moutot from 2023.
negative below the 1/3 threshold; the endpoint case is still open
Does the middle-third Cantor measure admit a Fourier frame, that is, a countable set of exponentials giving two-sided frame bounds on its $L^2$ space? No. The Cantor measure with base $b$ admits no Fourier frame for any odd integer $b > 1$, which answers Strichartz's question for the middle-third case.
For the integral $\mathcal{I}(h)$ over the unit interval and the torus, the paper gives the three-term Laurent polynomial $f(x,z)=(1-z^{-1})((1-x)+xz)$ with $\mathcal{I}(f^n)=0$ but $\mathcal{I}(z^{-1}f^n)=(-1)^{n-1}/(n+1)\neq0$. This disproves the $xz$-conjecture with one interval and one torus variable, shows $\ker\mathcal{I}$ is not a Mathieu–Zhao subspace, and by padding yields counterexamples for SU(2).
Wegner conjectured in 1965 that every finite family $\mathcal{R}$ of axis-parallel rectangles satisfies $\tau(\mathcal{R}) \le 2\nu(\mathcal{R}) - 1$, where $\tau$ is the minimum number of piercing points and $\nu$ the largest pairwise-disjoint subfamily. False, by an explicit triangle-free counterexample.
The Exact Matching problem asks whether a bipartite graph with edges colored red and blue admits a perfect matching with exactly $t$ red edges. Introduced by Papadimitriou and Yannakakis in 1982, it has been in randomized polynomial time since Mulmuley-Vazirani-Vazirani (1987) while membership in P stayed open for four decades. The paper claims a deterministic polynomial-time algorithm, replacing probabilistic amp…
Explicit counterexamples in dimensions 3 and 4, so GMC(n) fails for every n >= 3; GMC(1) was already known, and a separate human-authored preprint claims the remaining n = 2 case affirmatively
Can one construct a plain-model, efficient, information-theoretically secure one-time unclonable-encryption scheme for one classical bit with exponentially small adversarial advantage?
Maz'ya and Shaposhnikova introduced a non-classical maximal operator $M^\diamond$, the maximal convolution with the vector-valued signum kernel truncated to centered balls. One of Maz'ya's 75 open problems in analysis asks whether it can be separated from the sharp maximal operator $M^\sharp$. It can: there is a translation-invariant Banach space of locally integrable functions on which $M^\diamond$ is bounded but…
first counterexample of the form D^b(X) for X smooth projective
Simon conjectured that every skeleton of a simplex is extendably shellable. False: for every $d \ge 3$ there is a pure $d$-dimensional shellable simplicial complex that is not shelling completable.
integral unimodular lattices of rank at most 32; the general conjecture is open
Does bipartite bound information exist: classical correlations between two parties and an eavesdropper that cost secret bits to create, yet from which no secret key can ever be distilled?
Records, not resolutions: the exact values of these Ramsey numbers remain unknown.
Holevo and Werner's 1999 lower bound on the quantum capacity of the bosonic thermal attenuator comes from thermal inputs. Is it optimal? The paper proves it is exactly the supremum over single-mode Gaussian states, then exhibits a non-Gaussian state that beats it, giving positive quantum capacity in a region where every single-mode Gaussian input yields none.
The near-quadratic Elekes-Ronyai expander conjecture over $\mathbb{R}$ predicts that a nonspecial polynomial expands any finite set to near-quadratic size. False: a fixed nonspecial quadratic polynomial, together with arbitrarily large finite sets of real algebraic integers, has image with a fixed power saving from quadratic size.
For a projective variety $X$ with at worst Gorenstein canonical singularities whose stringy $E$-function $E_{\mathrm{st}}(X; u, v)$ is a polynomial, all stringy Hodge numbers $h^{p,q}_{\mathrm{st}}(X)$ are non-negative. (Batyrev 1998, Conjecture 3.10.)