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problems
If a chromatic symmetric function is Schur positive, must every finite-variable specialization X_G(x_1, …, x_k) have a saturated Newton polytope? A 12-vertex bipartite graph realizes weights (6,6,0) and (8,2,2) but omits their midpoint…
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.
For P_n(z) = ∑_k=0^n ε_k z^k with independent uniform signs, does the number R_n of roots in |z| ≤ 1 satisfy R_n/(n/2) → 1 almost surely? The manuscript proves the strong law with R_n = n/2 + O_ω(n^149/150).
Chromatic quasisymmetric functions of natural unit interval graphs were conjectured to have log-concave coefficients in the elementary basis. A connected 13-vertex example refutes it: for the Hessenberg function…
For deterministically minimizing a convex 1-Lipschitz function on the d-dimensional ball using only exact function values, the query complexity sat between Ω(d) and O(d^2 log^2 d) since 1996. The paper proves a near-quadratic lower bound…
Can the k-distinct language - words over [n] of length at most k with no repeated symbol - be recognized by an acyclic NFA of size c^k n^O(1) for some c < 4? A construction of size 2^1.96992k n^O(1) < 3.918^k n^O(1) answers yes.
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.
A subset A of the pointwise-ordered cube [0,1]^n is a k-antichain when it meets every chain in at most k points. The conjecture concerns the largest possible (n-1)-dimensional Hausdorff measure of such a set; it is settled here, following…
The Huneke–Wiegand Conjecture: Let R be a one-dimensional Gorenstein local domain, and let M be a finitely generated, non-zero, torsion-free R-module. If the tensor product M ⊗_R M^* is torsion-free, then M is a projective (hence free)…
For an arrangement of n hyperplanes in P^3_C with ℓ intersection lines and p intersection points where at least three hyperplanes meet, the refined form of Purdy's inequality expects p - ℓ + n + 2 ≥ 0. An explicit arrangement built from…
How many nondegenerate equilibrium points can the potential of three positive point charges have? Gabrielov, Novikov and Shapiro had proved at most 12, and observed that their method would give 6 if an auxiliary polynomial system had at…
Does every instance of indivisible goods with additive valuations admit a balanced allocation (any two bundles differing in size by at most one) that is simultaneously envy-free up to one good (EF1) and fractionally Pareto optimal (fPO)?…
Aldroubi, Cabrelli, Krishtal and Molter conjectured that for a bounded normal operator T and any vector g, the normalized orbit T^k g / |T^k g| : k ≥ 0 is never a frame. It can be: an explicit construction produces a normalized orbit that…
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…
Simon conjectured that every skeleton of a simplex is extendably shellable. False: for every d ≥ 3 there is a pure d-dimensional shellable simplicial complex that is not shelling completable.
VibeMathed records no statement for this problem. See erdosproblems.com for the original.
Englert and Rehacek conjectured which measurement is globally information-optimal for an ensemble of equiangular equiprobable pure states. Their conjecture holds, via the remaining entropy inequalities of Holevo and Utkin.
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)?