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Open problems, the work posted against them, and what checked that work.
problems
A collection of open problems drawn from published lists, including Cahen, Fontana, Frisch and Glaz's Open Problems in Commutative Ring Theory and Erman and Sam's survey of Boij-Soderberg theory, each proved or disproved by one automated…
For every δ > 0 and infinitely many n there is a set of n lines in the plane with no intersecting quadruple such that every subset of size at least n^4/5+δ contains three concurrent lines. This improves the bound for a dual form of a…
A problem from Fajtlowicz's Graffiti program, studied by Erdős and Staton, on the Havel-Hakimi residue of common-divisor graphs. The paper resolves the problem and extends it, determining the residue's first-order scale and its nontrivial…
Two degree inequalities for circle-valued Sobolev maps have constants that degenerate as p → 1^+ or δ → 0^+. Brezis posed the problem of sharpening them; both are now sharpened, by the same power trick with elementary estimates.
For every n≥2 the paper exhibits an n-dimensional K-polystable toric Q-Fano variety whose alpha invariant is exactly 2/2n+1, answering a question of Liu and Zhuang on whether a K-semistable example exists with alpha invariant between 1/n+1…
For the least k at which the small-prime part of binomnk exceeds n^2, how large can f(n) be?
For A ⊂ F_p of density 1/2, call A almost affine invariant under φ(x) = ax+b if |A triangle φ(A)| = o(p). Problem 90 asks for the threshold K below which A can be almost affine invariant simultaneously under all such φ with |a|, |b| ≤ K…
Tuza conjectured that every finite simple graph satisfies τ(G) ≤ 2ν(G), where ν counts pairwise edge-disjoint triangles and τ is the fewest edges whose deletion leaves the graph triangle-free. Puleo had proved it for maximum average degree…
Nazarov conjectured that for s ∈ (1, 3/2) the quadratic form of the spectral fractional Dirichlet Laplacian strictly increases under u ↦ |u| when u changes sign. Proved and substantially generalized, with the same conclusion for the…
Banach asked in 1932 whether a real Banach space X whose ndimensional subspaces, for some fixed 1 < n < dim X, are all isometric must be a Hilbert space. Gromov proved the conjecture for even n, and subsequent work settled several…
How few vertices can a triangulation of RP^5 have? The paper presents a 6-dimensional centrally symmetric simplicial polytope whose antipodal boundary quotient gives a 24-vertex triangulation, far below previous constructions in the…
If A generates a bounded C_0-semigroup on a Hilbert space and has dense range, does A^-1 also generate a bounded C_0-semigroup? Posed by deLaubenfels in 1988. Answered negatively: Lorist, Meyries and Veraar construct a bounded operator…
Let k≥ 3 and f_k(N) be the maximum of ∑_n∈ A1/n over all A⊆1,…,N containing no k subsets with the same pairwise least common multiple. Estimate f_k(N). The claimed answer: f_k(N)=(log N)^γ_k+o(1), where γ_k is a weighted generalization of…
Let T_k be the least t such that every equinumerous t-coloring of [tn] contains a rainbow k-term arithmetic progression. Jungic, Licht, Mahdian, Nesetril and Radoicic conjectured T_k = Θ(k^2); Conlon, Fox and Sudakov proved T_k = O(k^2 log…
Akbari, Alikhani, Oboudi and Peng conjectured in 2010 that 0 and -2 are the only integer roots of the domination polynomial D(G, x), proven for trees and unicyclic graphs and verified exhaustively for small orders. The paper gives a…
Seymour conjectured that every oriented graph has a vertex x with |N^++(x)| ≥ |N^+(x)|. It holds for oriented graphs of minimum out-degree exactly 7, the first improvement to the out-degree threshold since Kaneko and Locke settled degree 6…
Online Shadow Tomography with log m dependence, while retaining poly(log(d)/ε) dependence. Also, matching the best classical bounds for Adaptive Data Analysis
Crouzeix conjectured in 2004 that for every square complex matrix A and every polynomial p, lVert p(A)rVert ≤ 2 max_z ∈ W(A) |p(z)|, where W(A) is the numerical range of A - that is, the numerical range is a 2-spectral set. Crouzeix proved…
VibeMathed records no statement for this problem. See erdosproblems.com for the original.
How well separated can a family of point-line pairs in the unit square be? For every ε > 0 there are arbitrarily large families (x_1,ℓ_1),…,(x_n,ℓ_n) in [0,1]^2 with x_i ∈ ℓ_i and dist(x_i,ℓ_j) ≥ n^-2/3-ε for all i ≠ j. Combined with…