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Open problems, the work posted against them, and what checked that work.
problems
The interchange graph G(R,S) has the (0,1)-matrices with row sums R and column sums S as vertices, adjacent when they differ by a single 2× 2 interchange. Brualdi asked whether G(R,S) is always Hamiltonian. It satisfies more: it is…
For the Fubini numbers a(n), is a(n) = ∑_k=0^2^n-1-1 A284005(k) for every n > 0, as conjectured on the OEIS in 2018?
Every minimally generically globally rigid graph in R^d containing a subgraph isomorphic to K_d+2 is itself isomorphic to K_d+2, confirming Conjecture 6.3 of Garamvölgyi, Jackson and Jordán (2025).
WOW-284 asserts that the minimum dual degree of every connected graph of order at least three and girth at least five is at most the negative of its least distance eigenvalue. The paper refutes it with exact counterexamples of orders 38,…
For a semistable one-parameter family of complex projective varieties with smooth nearby fiber X_t and monodromy T, is the map H^1(X, Z) → H^1(X_t, Z)^T surjective? True in degree one, although the integral statement fails in higher degree.
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.