Problems
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Godara and Sarkar proved d(H_27)=6 for the exponent-p Heisenberg group and posed d(H_p^3)=3p-3 for every odd prime p, leaving p≥5 open. The paper settles the first open case, d(H_125)=12, the upper bound reducing to a finite spread bound…
We prove that every PPT linear map has finite entanglement-breaking index, thereby establishing the eventual entanglement-breaking property of PPT channels in full generality. Furthermore, by utilizing completely positive maps with low…
Energy measures of any two nonconstant harmonic functions on the standard Sierpinski gasket are mutually absolutely continuous. Strichartz and Tse reported numerical evidence that the Radon-Nikodym densities are L^p-integrable for 1 < p <…
Does planarity help approximate counting? The paper gives an FPRAS for the planar hard-core partition function at small activity, proves that approximately counting q-colourings on planar graphs is NP-hard for every constant q ≥ 4, and…
Sampling a nearly uniform Eulerian tour of a directed Eulerian multigraph was stuck at mn-type running times coming from arborescence sampling. A randomized algorithm achieves widetildeO(m^3/2) worst case, breaking that barrier on sparse…
Let H(n) be the largest number of vertices in a hypergraph with no isolated vertices and no partition of size greater than n. With k_1 = 1 and k_n = ⌊ n/2 ⌋ + k_⌊ n/2 ⌋ + k_⌈ n/2 ⌉, prove H(n) ≥ c k_n for some constant c > 1, already for n…
A t-(v,k,λ) covering is a family of k-subsets of a v-set meeting every t-subset at least λ times, and C(v,k,t) is the least number of blocks. The recorded bounds for C(12,6,4) were 40 ≤ C(12,6,4) ≤ 41. No 4-(12,6,1) covering with 40 blocks…
For a positive projection P on a Dedekind complete Banach lattice whose largest central operator below P is α id, Wickstead conjectured α must be 0 or 1/n for some natural n, and proved the finite-dimensional case. The paper proves the…
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…
VibeMathed records no statement for this problem. See erdosproblems.com for the original.
Escobar, Klein and Weigandt proved that gradedness of an ASM weak order interval, constancy of Coxeter length across its fibres, and equidimensionality of the associated ASM varieties are mutually equivalent, and conjectured (Conjecture…
Does every synchronizing one-cluster automaton on n states admit a reset word of length at most (n-1)^2? The new bound (m-1)(n-1) + mℓ ≤ (n-1)^2 settles the one-cluster case of the Černý conjecture.
The classes of SOP_2 and SOP_3 first-order theories coincide. This answers a question of Džamonja and Shelah from 2004.
VibeMathed records no statement for this problem. See erdosproblems.com for the original.
The imbalance of an edge uv of a finite simple graph is the absolute difference of the degrees of u and v. Kozerenko and Skochko conjectured that the multiset of all edge imbalances is graphic - realizable as the degree sequence of some…
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.
A finite closure system can be given by implications or by a list of subsets closed under intersection. Deciding whether one specification of each kind defines the same family had remained open in several settings; the paper proves the…
For the switch-walk-switch walk on Z_2 wr Z started at (0,0) and (0,2), prove |P_t^x - P_t^y|_TV asymp t^-1/2.
A collection of open problems from the algebraic and enumerative combinatorics literature, resolved in one paper: a conjecture of Defant, Jiang, Marczinzik, Segovia, Speyer, Thomas and Williams on the echelonmotion operator on modular…
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?