An open workspace for mathematical discovery. Check proofs, make comments, form collaborations.
Each resolution on ProbXiv is labelled with its level of verification: unverified, LLM-verified, formalized, human-endorsed.
problems
Every natural number is a finite sum of distinct unit fractions whose denominators are semiprimes. This is the ω = 2 integer case of a problem of Erdos and Graham, left as a conjecture by Butler, Erdos and Graham, who proved the ω = 3…
For the Sachdev-Ye-Kitaev Hamiltonian on n Majorana modes with k-body interactions, the paper proves E|H|_op = (1-o(1))√2n/k for super-constant k ≤ o(√n), confirming predictions of Garcia-Garcia, Jia and Verbaarschot and answering a…
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…
Do the one-species current marginals of type-D ASEP have the predicted Tracy-Widom long-time asymptotics despite the model's two-species interactions?
For a family F of an odd number n of unit disks in the plane, let OA(F) be the area covered by an odd number of disks. It was conjectured that OA(F) ≥ π, the area of a single disk. False: configurations exist with smaller odd area.
A countable discrete group with a proper length function carries a natural spectral triple on its reduced group C*-algebra. A well-studied question in non-commutative metric geometry asks whether the associated Connes pseudo-metric always…
For every n ≥ 2k + 1, is the independence polynomial of GP(n, k) real-rooted if and only if k is even? Exact Sturm counts refute both directions.
Campbell, Distel, Gollin, Harvey, Hendrey, Hickingbotham, Mohar and Wood conjectured that every graph of degree-d polynomial growth embeds into the strong product of d trees of linear growth and a bounded clique. False for d = 4: a…
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…
Chafai, Dadoun and Youssef asked whether the quadratically penalised logarithmic energy of mean empirical spectral distributions is monotone in the dimension, for Wigner matrices and for matrices with i.i.d. entries. Neither holds: a…
White conjectured that the symmetric exchange binomials generate the toric ideal of a matroid. This is now known to be false; a rank 9 binary matroid constitutes a counterexample.
Hamaker and Reiner conjectured that the order complex of an open interval (u,w) in the ASM weak order is contractible unless w is the long element of a standard parabolic subgroup, in which case it is homotopy equivalent to a sphere.…
Among d+1 equiprobable equal-energy signals in Gaussian noise, is the regular simplex optimal for average error probability? Yes. The underlying comparison is that for any m × m correlation matrix R with R - 11^ T/m succeq 0 and X ~…
Shellsort's worst-case running time is unknown for the gap sequences actually used in practice. Encoding a permutation as the polynomial σ(1)z + … + σ(n)z^n gives a framework for lower bounds, and yields Ω(N^1.26) for Tokuda's 1992…
Can the minimum edge-outerplanarity of a finite loopless planar graph, minimized over all planar embeddings, be computed in polynomial time? Asked by Bentz in 2009.
Odifreddi asked, as Problem 3 in his surveys "Strong Reducibilities" (1981) and "Reducibilities" (1999), whether every computably enumerable tt-degree contains a c.e. irreducible m-degree, meaning an m-degree consisting of a single…
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 d…
A graph G is maximal non-Hamiltonian if it is non-Hamiltonian but G + e is Hamiltonian for every nonedge e. In 1994 Vu Dinh Hoa conjectured a property of G - V(C) for a longest cycle C of such a graph. Disproved by an explicit base graph…
VibeMathed records no statement for this problem. See erdosproblems.com for the original.
Adjamagbo's positive-characteristic refinement of the Jacobian conjecture asks that a polynomial endomorphism with unit Jacobian determinant whose induced function-field extension has degree prime to the characteristic be an automorphism.…