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The Kajitani–Ueno–Miyano conjecture asserts that every finite uniformly dense matroid has a cyclic basis ordering. The conjecture is proved for all matroids of rank three. The new result establishes the previously unresolved divisible…

For an even cycle of size N and depth p with 2p + 2 ≤ N, is the optimal QAOA approximation ratio for MaxCut exactly 2p+1/2p+2, as Farhi, Goldstone and Gutmann conjectured?

VibeMathed records no statement for this problem. See erdosproblems.com for the original.

Is the closest vector problem NP-hard to approximate within polynomial factors n^c? Yes for some c > 0: hardness of approximation reaches polynomial factors, with consequences for decoding and related lattice problems - a foundational…

In the circle of Kummer's regular primes and Vandiver's conjecture, the paper proves that almost all primes are partially regular, yielding a partial Vandiver theorem for a density-one set of primes, with consequences for Kubota-Leopoldt…

VibeMathed records no statement for this problem. See erdosproblems.com for the original.

VibeMathed records no statement for this problem. See erdosproblems.com for the original.

Grothendieck asked whether every finite locally free group scheme of order n is killed by n (its n-th convolution power map equals the unit). The counterexample is an order-4 group scheme not killed by 4 (killed only by 8); since Deligne…

For triangular arrays of nodes a_i^n∈[-1,1] let L^nf be the Lagrange interpolation polynomials of a continuous f, with fundamental polynomials p_i^n. Is there a choice of nodes such that for every continuous f there is some x where…

If f(n) is the maximum total side length of n interior-disjoint squares packed in the unit square, is f(k^2 + 1) = k? An exact rational configuration packs 17 squares with total side length greater than 4, refuting the identity at k = 4.

Huang, Jiang and Oblomkov conjectured that the Eulerian q-series counting commuting pairs of nilpotent matrices with X^a = Y^b equals an explicit theta-and-eta product, making the point count essentially modular. The conjecture is layered…

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,…

Let p be a complex polynomial of degree n ≥ 2 whose zeros all lie in the closed unit disk. Then for every zero a of p, there exists a critical point ζ of p such that |ζ-a| ≤ 1. This is the standard Sendov statement and exactly matches the…

A generalization of Boppana's entropy inequality, of the kind used in union-closed-sets arguments, proved and formalized: the sharp form with the extremal constant characterized via the unique positive solution of an explicit equation.

VibeMathed records no statement for this problem. See erdosproblems.com for the original.

For smooth convex-concave min-max problems, can anchored gradient descent-ascent be scheduled so that its exact last-iterate squared-gradient residual is O(1/t), closing the gap left by the 2019 analysis?

If n_1 < n_2 < … with n_k+1/n_k ≥ c > 1, must ∑_k 1/F_n_k be irrational? The proposed proof closes the range 1 < c < 2 left open by earlier criteria.

VibeMathed records no statement for this problem. See erdosproblems.com for the original.

VibeMathed records no statement for this problem. See erdosproblems.com for the original.

Is every weakly quasi-complete Noetherian local ring quasi-complete? Asked by D. D. Anderson in 2014. The ring A = k^p[[X, Y]][k] with k = F_p(u_1, u_2, …) is weakly quasi-complete but not quasi-complete.