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exponent 2 fails for every p > 2; the sharp exponent p' form is proved for integer p ≥ 2
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.
density-one set of n; the conjecture itself remains open
After Chen-He-Ye-Yuan's counterexample to direct three-block ADMM, the subclass in which the third constraint block is the identity matrix remained unresolved: the literature contained neither a convergence proof nor a counterexample. Disproved: an explicit rational counterexample exists in which the first two blocks are strongly convex quadratics and direct three-block ADMM produces a bounded nonconvergent orbit of…
up to polylogarithmic factors
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.
the omega = 2 integer case, the one Butler, Erdos and Graham left open
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 five-dimensional case of the Geode challenge of Amdeberhan, Kauers and Zeilberger, concerning the geode factor attached to a family of multivariate generating functions. Settled in dimension five.
Does the Benjamini-Hochberg procedure always control the false-discovery rate at its nominal level for correlated two-sided Gaussian p-values? A factor model gives at nominal level .
The sharp valuation bounds and optimal are proved for ALL loop quivers , hence for the extremal BPS invariants of all twist knots (both rows, matching every twist-knot entry of GKS Table 1). Scope limits: the /figure-eight divisibility was previously proved by Basor–Conrey–Morrison (arXiv:1703.00990), whose per- -adic characterization for is finer than the uni…
The first rigorous solid standard Young tableaux challenge asks for a proof of a conjectured second-order recurrence for the number of solid standard Young tableaux. The conjectured recurrence is proved.
settles two numbered Erdos problems at once
For the Erdős–Pomerance functions and counting how many consecutive integers are needed to contain a distinct multiple of each integer, respectively prime, up to , the paper proves , disproving Kominers' conjecture that . The paper also significantly improve…
The manuscript claims is transcendental for every and , settling the positive integer-valued affine subclass of Erdős Problem 270. Two pieces of context matter. Problem 270 as Erdős and Graham posed it, for every , was already answered no by Crmarić and Kovač in 2025: for any some such makes the series sum to . What survives is the non-decreasing case,…
Improved bounds rather than a settled question: the asked-for inequality is not established in general.
A precise asymptotic formula for the number of partial Hadamard matrices in the regimes and , reaching the cubic regime that previous approaches (de Launey-Levin and successors) could not.
Ekhad and Zeilberger's second computational Chomp challenge asks for a Chomp position with three winning opening moves. Answered by exhibiting a bar with three winning opening moves.
One conjecture each way: the second proved, the first disproved. Two further Chen-Lawrencenko conjectures remain open and are flagged as such in the paper.
A polynomial-time algorithm for computing an optimal committee under any Thiele voting rule on the Voter Interval domain, resolving a ten-year-old open problem posed for Proportional Approval Voting by Elkind and Lackner and later extended to every Thiele rule.
Among classes of tournaments for which neither hardness nor polynomial-time solvability of isomorphism was known, bounded VC dimension stood out as an open problem of Neuen and Grohe. Resolved: isomorphism of tournaments of VC dimension is decidable in time , so automorphism groups of bounded-VC tournaments are computable in polynomial time; isomorphism of tournaments of bounded chromatic number…
For a positive projection on a Dedekind complete Banach lattice whose largest central operator below is , Wickstead conjectured must be or for some natural , and proved the finite-dimensional case. The paper proves the conjecture in general and settles the representation problem for Banach lattice algebras as a consequence.
Wellman and Pettie noted that the true leading constant for large-order Davenport-Schinzel sequences was known only to lie in an interval. The paper improves the Roselle-Stanton lower bound to match the pigeonhole upper bound in the leading term, resolving the constant to exactly 1/2.
Question 6.1 of Chalmoukis, Tsikalas and Yakubovich asks how far the Power boundedness constant of a matrix can exceed its ordinary Kreiss constant . Answered more strongly: for every there are matrices whose Cayley transforms satisfy while the strong Kreiss constant satisfies with . Since…
The manuscript claims a polynomial-time reduction from 3-SAT proving -C1P NP-hard; together with membership in NP, this establishes NP-completeness and closes the sole unresolved case from the earlier classification. It also implies NP-completeness of the equivalent completion problem. The proof package further shows that, within its specific nested-prefix/internal-local gadget architecture, no…
the PIR consequence is conditional on a number-theoretic conjecture implied by either the generalized repunit conjecture or Schinzel's hypothesis H, and is unconditional for s <= 15
Subbarao and Verma asked in 1999 (Problem 5.7, first part) whether the complementary Bell numbers take any given value only finitely many times. Campbell proves they do: for every fixed integer the fiber is finite, a result whose techniques connect to Wilf's conjecture on the vanishing of .
The paper also proves the conjecture in the unimodular case and characterizes equality there, so the boundary between true and false is drawn rather than just crossed.
For the adjacent-transposition chain on with a regular parameter vector, Fill's spectral gap conjecture (recently resolved) leaves open the characterization of the equality cases. The paper settles them, constructing the additional eigenfunctions in the exceptional regime.
An improved lower bound on the maximal product; the sharp maximizer for the 1958 question remains unknown.
Klopp and Zadik gave an exponential-time node-private algorithm for exact community recovery in stochastic block models and asked whether a polynomial-time algorithm could match it. One can: a Lipschitz surrogate for the penalized likelihood plus an accept-reject sampler gives a high-probability polynomial-time node-private algorithm that nearly matches the exponential-time guarantee.
Arjevani et al. asked whether almost-surely bounded oracle error permits a better rate than bounded variance for smooth nonconvex stochastic optimization. It does not: every randomized adaptive algorithm still needs Omega(dL/eps^2 + dL sigma^2/eps^4) queries, matching the standard upper bound.
Improves every prior result for p >= 2 and matches the classical extragradient method at p = 1.
The Foregger–Sinkhorn tie-point conjecture, Conjecture 41 in Minc's survey, asserts that if a nearly decomposable doubly stochastic matrix minimizes the permanent on a face and the permanental cofactor at a prescribed zero exceeds its permanent, then that zero is a tie point. False: an explicit counterexample exists, built on the unique root of in .
Akbari, Alikhani, Oboudi and Peng conjectured in 2010 that 0 and -2 are the only integer roots of the domination polynomial , proven for trees and unicyclic graphs and verified exhaustively for small orders. The paper gives a counterexample of order 33 with an integer domination root at , built from an S-unit branch cancellation mechanism.
Pełczyński's duality between strictly singular and strictly cosingular operators fails without weak compactness. Beanland asked, in work with Androulakis and later on MathOverflow, for the separable-range case: the paper answers it affirmatively and shows that for separable , is strictly cosingular exactly when is strictly singular.