Problems
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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…
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.
Does a universal summation process recover the degree of a circle map from its Fourier moduli, that is, does ∑_n σ_n,ε n |hat f(n)|^2 → deg f hold for Holder maps below the threshold? No. For every 0 < α < 1/3 there is an f ∈…
For the three-dimensional paraboloid P_3 over a prime field in which -1 is not a square, the Fourier extension operator maps L^2 to L^r for r > 176/51 = 3.45098…, improving the exponent by combining a bilinear approach with point-line…
New upper and lower bounds on the approximation ratio achievable for Multiway Cut via large mixtures of new and old rounding schemes for the CKR relaxation, advancing the ratio ladder that has run since Călinescu-Karloff-Rabani (1998).
Let A(k) be the largest possible number of moves in a north-east lattice path whose visited vertices contain no k collinear points. Gerver (1979) and Gerver and Ramsey (1979) bounded A(k) by exp(Ω(log(k)^2)) ≤ A(k) ≤ exp(O(k^4)), and…
Korsky, Saffat and Aiylam bounded the growth constant c(G) for integer-valued Lipschitz functions on G(n,d/n) between 1/(2d) and 4log^2 d/d up to lower-order terms. The random-graph side is sharpened.
Let n_k be the least integer greater than 2k for which ∏_i=1^k (n_k - i) has no prime factor in (k, 2k). How rapidly must n_k grow?
Vinzant conjectured, in a form later restated by Bandeira, that the 4M-4 threshold for injective complex phase retrieval is sharp. Part (1) holds: for A ∈ C^N × M with N = 4M-5 and i.i.d. standard complex Gaussian entries, the phase…
How well can an arbitrary boolean constraint satisfaction problem of arity k be approximated in polynomial time? The paper gives a (k/2^k)-approximation, improving the previous best constant of 0.626612 k/2^k due to Makarychev and…
Is the fractional chromatic number of every d-degenerate triangle-free graph at most (1+o(1))d/log d, with a matching lower bound, as conjectured by Martinsson and Steiner? The upper bound is confirmed constructively for graphs of girth at…
Let g(r) be the fewest edges in an r-uniform intersecting hypergraph with cover number r. Erdos and Lovasz proved g(r) ≥ 8r/3 - 3. An elementary argument gives g(r) ≥ 3r - 4, and building on it with Kahn's small-codegree edge-colouring…
The paper nearly settles the tradeoff between the size of a set system over [n] and its number of full chains, an extremal question raised by Johnson, Leader and Russell as a counterpart to Sperner-type results, and linked by recent work…
Near-optimal density thresholds forcing a measurable set in R^d to contain all sufficiently large similar copies of every n-point configuration, answering a question from the Euclidean density theorem literature up to logarithmic factors.
The Courtade-Kumar conjecture (2014) posits that dictatorship functions maximize mutual information between a Boolean function's output and a noisy input. The paper resolves an open question posed by Courtade and Kumar themselves - a sharp…
The Lonely Runner Conjecture of Wills and Cusick states that among k+1 runners at distinct constant speeds on a unit circle, each runner is at some time at distance at least 1/(k+1) from all others. Following Rosenfeld's computer-assisted…
Pavez-Signe (2024) conjectured a Dirac-type condition for spanning H-subdivisions and asked whether the subdivision paths can additionally be required to have similar lengths; Lee (2025) resolved the existence conjecture in the stronger…
Regev and Stephens-Davidowitz conjectured that Z^n maximizes the Gaussian mass Θ_L(t) = ∑_x ∈ L e^-t|x|^2 over stable lattices for every t > 0. The sharp inequality holds for every integral unimodular lattice of rank n ≤ 32, with equality…