Problems
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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…
Does every globally asymptotically stable homogeneous polynomial vector field admit a homogeneous polynomial Lyapunov function? No. A planar homogeneous cubic vector field with integer coefficients is globally asymptotically stable yet…
Can one hear the shape of a drum, in the Steklov setting and in the plane? No: there exist pairs of noncongruent bounded plane domains with identical Steklov spectra including multiplicities, simply connected, strictly convex, with…
Is the difference between the numbers of positive and negative adjacency eigenvalues of every connected line graph at most one? A 14-vertex witness has signature 2, and chaining copies gives connected line graphs of signature k + 1 for…
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.
Does every nontrivial finite simple graph have noninteger Sombor energy? If ρ_1,…,ρ_n are the eigenvalues of the Sombor matrix of a graph G, its Sombor energy is E_SO(G)=∑_i=1^n|ρ_i|. The conjecture asserted that E_SO(G)∉ Z for every…
Can a finite set of lattice points determine many rectangles but few isosceles triangles? Both parts of the governing question have negative answers, quantified by explicit blowup rates, and the resulting configurations give obstructions…
The classes of SOP_2 and SOP_3 first-order theories coincide. This answers a question of Džamonja and Shelah from 2004.
For s ∈ (1/4,1) and any degree, the only W^s,1/s-minimizers among maps S^1 → S^1 are Blaschke products. This resolves Open Problems 23 and 24 of Brezis and Mironescu's book on mappings to the circle, and Brezis's Favorite Open Problem 5.4…
The kissing number in 19 dimensions is at least 11948, improving the Cohn-Li bound by 256, via a binary code of length 19 and minimum distance 5 fed through the Cohn-Li odd-sign construction.
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 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…
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…
A conjecture attributed to Kontsevich holds that strata of quadratic differentials are aspherical, that is orbifold K(π,1) spaces. False: when there are at least four zeros or poles, no connected component of a genus-one stratum is an…
Fulek defined a weight-five three-row 0-1 matrix L_3 and asked whether ex(n, L_3) = O(n). It is: every r × s matrix avoiding L_3 has at most 27r + 2s ones, so 6n - 8 ≤ ex(n,L_3) ≤ 29n for n ≥ 5. The same argument covers an infinite family…
Can S-decoding polynomials modulo a product of k primes be built with only k+1 nonzero coefficients, the minimum their own lower bound allows? Yes, via a general framework for special prime products. The consequence is that for any…
Improved lower bounds for nine classical Ramsey numbers, including R(3,13) ≥ 61, R(3,18) ≥ 100, and seven R(4,k) records up to R(4,20) ≥ 237, found by AlphaEvolve-discovered search algorithms.
Does quantum memory give a query-complexity advantage for learning an unknown quantum channel, when protocols without it must measure after each channel use and keep only a classical transcript? It does, and the paper also determines how…