Problems

The Howland-Kato conjecture that every nonzero positive commutator i[f(P),g(Q)] must arise from functions in appropriate Kato classes is false: i[arctan(P),arctan(Q)] is nonzero and nonnegative.

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

Is the depth of the mod-p cohomology ring of every finite group realized as the dimension of one of its associated primes? For G = SmallGroup(128, 859) over overlineF_2 the ring has depth 2 while every associated-prime quotient has…

Does the natural trace estimate hold for kinetic energy spaces in the unrestricted Gaussian velocity model on bounded domains (Question 1.8 of Albritton, Armstrong, Mourrat and Novack)? No: for each 1 ≤ p < 2 there are counterexamples on…

If a finite graph has girth at least five, must its minimum dual degree satisfy δ^*(G) ≤ -∂_n(G), where ∂_n(G) is the smallest eigenvalue of its distance matrix? The Hoffman-Singleton graph violates it: dual degree 7 against eigenvalue…

Erdős and Graham asked whether binomnk with 1 ≤ k ≤ n/2 must always have a divisor ≤ n that is close to n, meaning bigger than a fixed constant times n. Settled in both directions: true when k is large enough as a function of n, but false…

Monical, Tokcan and Yong conjectured that Schubitopes, the generalized permutahedra arising as Newton polytopes of Schubert polynomials and of Demazure characters of GL_n, are Ehrhart positive. Disproved by an explicit Schubitope whose…

For a finite abelian group G, let Φ(G) be the absolutely convex hull of the specified trilinear kernels and Φ'(G) its restriction where the third factor depends only on x_1 + x_2. Is Φ(G) = Φ'(G)? A counterexample over Z/3Z separates the…

For a bipartite graph without isolated vertices and with a unique minimum dominating set, does the proposed function m(n, γ) bound the number of edges whenever γ ≥ 2 and n ≥ 3γ? A 13-vertex bipartite graph with 22 edges exceeds the…

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

Huybrechts conjectured that for every Brauer class alpha on a hyperkahler variety X, the index divides the period raised to the power dim(X)/2, strengthening the usual period-index conjecture. Disproved on certain hyperkahler fourfolds, in…

For p ≥ 2, does Carbery's proposed many-function almost-orthogonality inequality hold with the pairwise overlap coefficients raised to the power 2 - and if not, what is the largest possible exponent?

Lassak conjectured that a reduced planar convex body of thickness Δ has area at most (π/4)Δ^2, the value for the disc. False: an explicit reduced body of thickness 1 has area 0.786215… > π/4 = 0.785398…, given by a closed-form support…

Mihail and Vazirani conjectured that the graph of every 0/1-polytope has edge expansion at least one. Disproved by a family of 0/1-polytopes whose edge expansion decreases exponentially in the dimension.

Friedland and coauthors proposed a quantum analogue of the p-Wasserstein distance and conjectured that, though only a semidistance in general, it is a true distance for a particular quantum cost matrix and for cost matrices near it. The…

Sabok asked whether the compact convex set S'(X) attached to a separable metric space of diameter at most one is always a simplex, and whether S'(U_1) is the Poulsen simplex. Both answers are negative, with obstructions already visible for…

Kusner conjectured in 1983 that the maximum number of points in R^n that are pairwise at ℓ_p-distance one is exactly n+1 for every 2 < p < ∞, as in the Euclidean case. False: an explicit configuration of n+2 equilateral points exists for…

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

Conjectured upper bound on how many pairs among n points in the plane can be exactly one unit apart.

Must every connected graph satisfy the proposed upper bound on its independence number in terms of residue and largest induced-bipartite-subgraph order? The family overlineK_2r+1 ∨ (K_r sqcup K_r) violates it for every r ≥ 3.