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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.
How large can a measurable be while avoiding the vertices of upward-oriented axis-aligned right triangles of area ? At most , with a matching-shaped lower bound construction.
A finite closure system can be given by implications or by a list of subsets closed under intersection. Deciding whether one specification of each kind defines the same family had remained open in several settings; the paper proves the problem coNP-complete.
Reading computed the order dimension of the poset of regions for most finite Coxeter arrangements, observed that an exceptional type whose dimension exceeds its rank would be the first known simplicial arrangement with that property, and recorded the general guess that every simplicial region poset has dimension equal to its rank (Problem 9.3 of his 2016 chapter); Segovia later asked the analogous question for orien…
The quartet distance counts the four-leaf subsets on which two binary phylogenetic trees display different topologies. Bandelt and Dress conjectured the maximum over trees on leaves. Proved: it is , by reducing arbitrary pairs of trees to caterpillars through a common-root planarization and an identity on five-leaf trees.
Conditional on the randomized exact-volume Small-Set Expansion Hypothesis, and stated for least-squares objectives rather than sparse convex optimization in general.
Three immediate consequences follow for undirected unweighted planar graphs: better compression of the Okamura-Seymour metric, less space for constant-time exact distance oracles, and a faster distributed algorithm.
Is the depth of the mod- cohomology ring of every finite group realized as the dimension of one of its associated primes? For over the ring has depth while every associated-prime quotient has dimension at least .
Moraga conjectured, and Kollár and Zhuang recorded, an odd-dimensional extension of the rank bound for faithful abelian -group actions on smooth Calabi–Yau varieties. The paper disproves it.
two of the three remaining cases; one is still unclassified
Is computing a Kemeny-optimal aggregate ranking NP-hard when the input consists of exactly three complete rankings? Hardness was known for every even ; three voters was the minimal open case, and is polynomial-time solvable.
Teschner's universal bound b(G) <= (3/2)Delta(G) is false: the 18-vertex cubic bipartite graph has b(G) = 5 against a bound of 4.5. What survives is the restricted statement Teschner actually proved, that the bound holds for graphs of domination number at most three, and Gagarin and Zverovich's 2013 result that it holds for almost all graphs. The counterexample does not suggest a replacement bound, and the correct g…
part (1) of the conjecture
the omega = 2 integer case, the one Butler, Erdos and Graham left open
Boots and Royle, and independently Cao and Vince, conjectured that the join of an edge with a path on vertices is the unique planar graph of maximum adjacency spectral radius for every . Tait and Tobin proved it for sufficiently large in 2017; the conjecture now holds for all .
Removes the constant factor from the previous best guarantee; whether is optimal is not settled here.
Given planks of fixed total width, how should they be placed to cover as much of a convex body's volume as possible? Karoly Bezdek asked whether, for a Euclidean ball, the optimum is a single plank centred at the origin. It is, and the paper also settles every planar convex body.
The Hessian conjecture asks whether every polynomial with has a polynomial gradient inverse. It is known for , false for , and open exactly in dimension four, where it implies the plane Jacobian conjecture. Proved for every quartic polynomial in dimension four: the quartic case reduces to wi…
Mauri and Moraga posed a two-part question about log Calabi-Yau pairs whose boundary decomposes into big divisors. Both parts have negative answers.
Do a finite group's order together with determine whether the group is simple? A simple and a non-simple group of order share the statistic .
Englert and Rehacek conjectured which measurement is globally information-optimal for an ensemble of equiangular equiprobable pure states. Their conjecture holds, via the remaining entropy inequalities of Holevo and Utkin.
What is the optimal competitive ratio for online vertex cover when edges arrive one at a time? The paper proves a tight factor-2 lower bound via a reduction in the blueprint framework of Assadi, Jiang and Xiang, closing the gap left by prior work.
pins the sharp threshold; the conjecture itself was already known false above dimension two
Online Shadow Tomography with dependence, while retaining dependence. Also, matching the best classical bounds for Adaptive Data Analysis
A question of Averkov, Hofscheier and Nill on whether the Ehrhart -polynomial of a lattice polytope of large lattice width is real-rooted. Proved in fixed dimension for sufficiently large lattice width, giving strict log-concavity and unimodality of the -vector, with the analogous statement for the local -polynomial of a lattice simplex.
Determine the leading asymptotic of the largest eigenvalue of the -Majorana quartic SYK Hamiltonian as . The preprint proves almost surely, via the limiting free energy at every fixed positive temperature.
disproved at d = 4; the conjecture for smaller d is untouched
Proves general cases of the conjecture rather than every case.
Is the O'Shea-Zames-Falb multiplier test necessary for robust stability of Lur'e systems with slope-restricted nonlinearities, as conjectured by Carrasco? No: there is a stable Lur'e interconnection, certified by a full-block multiplier, that admits no OZF multiplier.
Dimension 5 only; public AI-generated candidate with no independent specialist review.
Makeev conjectured it to be true for all dimensions. This result disproves it for dimensions 4 and 5. Dimensions 6 and above remain open.
Monical, Tokcan and Yong conjectured that Schubitopes, the generalized permutahedra arising as Newton polytopes of Schubert polynomials and of Demazure characters of , are Ehrhart positive. Disproved by an explicit Schubitope whose Ehrhart polynomial has a negative coefficient in its monomial expansion.
Swinnerton-Dyer (1981) proved -equivalence trivial on smooth cubic surfaces over -adic fields with good reduction, except for three special types. The paper resolves two long-standing exceptional cases: triviality for the diagonal cubic over , answering a question from Manin's Cubic Forms (1972), and the cubic with universal equivalence of exponent 2 (Kanevsky, 1982).
Can online vector balancing in the Spencer setting achieve the optimal order of prefix discrepancy with an efficient algorithm?
In the list update problem, is the simple transposition rule optimal under IID requests? The question traces to Rivest's 1976 study of self-organizing lists. The paper proves transposition is within a small constant factor of the optimal online algorithm under any IID distribution.
Can the edges of a finite connected multigraph, given a closed eulerian trail, be partitioned into circuits so that no circuit contains two edges used consecutively in the trail? The proof in fact four-colours the edges to satisfy the constraints.
Erdos and Graham asked whether a positive-density subset of can avoid having any two distinct elements whose unit fractions average to a unit fraction. It can: there is a constant such that for all large some of size has that property, which also gives the best known lower bounds for related unit-fraction avoidance problems.