Problems
Everything in the archive: the problem as it was posed, what has been attempted against it, and who checked each attempt. The mark down the left of the list says who has looked — a person, a machine, or nobody yet. Human reviews and machine checks are counted separately and are never added together.
91 problems
How large must arithmetic circuits and formulas computing the n × n permanent be? New lower bounds include an arithmetic-formula bound of order n^4/log n, far beyond the quadratic barrier that stood for decades.
How dense can a sphere packing in R^n be as n → ∞? The Kabatiansky-Levenshtein upper bound stood for almost fifty years; the new proof improves the asymptotic upper bound all the way down to the Cohn-Elkies linear-programming threshold.
Let A(n) be the least positive integer not dividing binom2nn. Erdos asked for the behaviour of A(n) for reasonable n. Under an explicit dyadic-regularity formalization of reasonable, the distribution is determined on dyadic intervals…
For n ≥ 4, the natural scalar Poisson-summation certificates cannot prove the Regev-Stephens-Davidowitz Gaussian mass conjecture: any such certificate saturates, so the whole approach is blocked.
The Kajitani–Ueno–Miyano conjecture asserts that every finite uniformly dense matroid has a cyclic basis ordering. The conjecture is proved for all matroids of rank three. The new result establishes the previously unresolved divisible…
Erdos and Hajnal asked whether h_r(G) = maxχ(H) : H ⊆ G, girth(H) ≥ r tends to infinity as χ(G) does, for every fixed r ≥ 4. It does in every fixed polynomial edge-density regime.
Kotzig conjectured that for every even n ≥ 4 the complete graph K_n decomposes into n-1 perfect matchings such that every pair of them forms a Hamilton cycle. An asymptotic version holds: K_n decomposes into n-1 perfect matchings of which…
Can an S-decoding polynomial modulo a suitable product of k primes attain the lower-bound minimum of k + 1 nonzero coefficients? A construction matches the bound for special products of k primes, yielding exponentially fewer-server PIR.
Dittert's conjecture asserts that among nonnegative n× n matrices whose entries sum to n, the functional φ(A)=∏_i r_i+∏_j c_j-per(A) is uniquely maximized by J_n/n. The paper proves the case n=16 which, with Pang's result for n≥17,…
Does there exist a good pairwise-coprime sequence u_n with ∑ 1/u_n < ∞ and polynomial growth? What if one only requires u_n ≤ e^o(n)?
Anari, Charikar and Ramakrishnan asked whether every fractional perfect matching admits a perfect matching that is α-thin with respect to it, meaning it crosses every cut at most α times the fractional amount. Resolved up to…
Huang, Jiang and Oblomkov conjectured that the Eulerian q-series counting commuting pairs of nilpotent matrices with X^a = Y^b equals an explicit theta-and-eta product, making the point count essentially modular. The conjecture is layered…
R(c) = 40c+41 for every c ≥ 2 such that c+1 is divisible by 3, 4, 5, or 7 (covering ≈ 66% of all c); the full conjecture (Myers 2015 Conj. 4.9, ABEMRS16 §5.5) reduces to prime cases p ≥ 89, all smaller primes settled by SAT. Twenty-eight…
Espuny Diaz, Lichev and Wesolek conjectured that a Dirac-type minimum degree condition forces Hamiltonicity in spanning subgraphs of cycle powers. Asymptotically true: for every ε > 0 and all large k, any spanning subgraph of the kth power…
Carefully designed stepsize schedules alone accelerate plain gradient descent beyond its textbook O(1/T) rate, without momentum. Whether they can reach the optimal O(T^-2) was open. A lower bound of Omega(T^-1.9319) for last-iterate…
Extending the minimal model program beyond threefolds in positive characteristic is a standing goal of birational geometry. Assuming the log resolution conjecture for all log pairs birational to X, the cone theorem holds for projective log…
Djament asked whether a Grothendieck category satisfying suitable finiteness and exactness conditions must be equivalent to a module category. In the locally noetherian case the answer is no: there is a Grothendieck category with a…
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.
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.