Problems
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Ballantine, Beck, Feigon and Maurischat introduced the subsum polynomial sp(λ,x) := ∏_i (1+x^λ_i) attached to an integer partition λ, studied rational functions built by summing reciprocals of these polynomials over natural classes of…
How large can a Bruhat interval in S_n that is a poset hypercube be? Using a permutation pattern suggested by AlphaEvolve, the authors exhibit hypercube intervals of dimension O(n log n) for n a power of 2, matching the largest possible…
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…
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…
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…
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…
Can a complete edge-coloured, complex-weighted graph realize perfect-matching amplitudes of one on every monochromatic inherited vertex colouring and zero otherwise? Nonexistence is proved in the diagonal family N = D for every even N ≥ 4,…
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…
Twisted Deligne products categorify the tensor product of two Grothendieck rings. Classifying them leads to categorical n-cocycles, and Johnson-Freyd, Ostrik and Yu asked whether these are always pullbacks of ordinary group cocycles on the…
In the all-heads coin game a player starts with n coins, each showing heads with probability p; each round all remaining coins are flipped, the player must set aside at least one head (losing if none shows), and wins once all coins are set…
Let M(n) be the supremum of ∑_a ∈ A 1/(n-a) over pairwise coprime A ⊂ [1,n). Erdos asked whether M(n) ≤ ∑_p<n 1/p + O(1) uniformly. The average order is settled: ∑_n ≤ N M(n) = e^-γ N loglog N + O(N).
Pach conjectured that n Jordan arcs, pairwise crossing exactly once with no triple points, have O(n) tangent pairs. The best known bound stood at O(n^7/4); the paper improves it to O(n^3/2) (and O(n^5/3) in the at-most-one-crossing…
What is the maximum size of a binary code of given minimum distance? The linear-programming bounds of McEliece, Rodemich, Rumsey and Welch (1977) resisted improvement for half a century. The new upper bounds are exponentially stronger at…
Whether negative-partial-transpose states undistillable from one copy become distillable from finitely many copies is a basic open problem in entanglement theory. In the canonical two-parameter DiVincenzo family used as its…
The dimer constant of Z^3, the exponential growth rate of perfect matchings of the cubic lattice, has no closed form and is pinned only by bounds. The upper bound improves from Lundow's 0.457547, standing since 2001, to 0.452130, via…
For an infinite planar set in strong general position, how large can the chromatic and clique numbers of its positive-integer-distance graph be - in particular, can the chromatic number be infinite? Yes: there is such a set, no three…
Deng, Tidor and Zhao asked whether [N] admits a coloring with N^o(1) colors and no symmetrically coloured 4-term arithmetic progression, giving an O(N^log_223) coloring. The paper gives an O_k(N^4/k^2) coloring of [N] avoiding…
Reiner conjectured a description of the homotopy types of intervals in higher Bruhat orders. In corank 3 it holds: the facial intervals of B(n,n-3) are exactly the spherical intervals, and every other interval is contractible.
For the least cutoff c(n) after which every k occurs as the number of homothetic cubes in a decomposition of the unit n-cube, is c(n) ≫ n^n? The Lean proof shows c(n) = o(n^n) along odd dimensions.
Determine the Shannon capacities of odd cycles beyond C_5, or improve the best explicit bounds. Lovasz's theta function settled C_5 in 1979 and every longer odd cycle has stayed open since. The current records, all obtained with model…