Problems

Araujo, Piga and Schacht asked whether density and codegree both above 1/4 force a tight Hamilton cycle in a linearly quasirandom 3-graph. No: the threshold is p_0 = max_0 ≤ x ≤ 1minx^3, 1-x ≈ 0.3177, and below it there are dense 3-graphs…

Han and Jiang asked whether being of klt type is an open condition in flat families of varieties. It is not.

The low-degree conjecture predicts that when the low-degree advantage between a planted distribution and a uniform null distribution stays bounded, no polynomial-time algorithm can distinguish them. It is false. There is a planted…

Sivaraman asked whether perfect divisibility is characterized by its chromatic consequence: is a graph G perfectly divisible if and only if χ(H) ≤ binomω(H)+12 for every induced subgraph H of G? False: the Paley graph P(17) satisfies the…

For a finite-dimensional algebra A, finite global dimension forces HH_n(A) = 0 for all large n. Han conjectured the converse: eventual vanishing of Hochschild homology should detect homological smoothness. Disproved by an explicit…

For a forbidden configuration F, the Anstee-Sali conjecture predicts that forb(m, F) is Theta(m^(X(F)-1)), where X(F) comes from an explicit product construction. Disproved: the 4-uniform family on six vertices formed by a two-vertex core…

Let S(x) count ordered pairs (a,b) with a+b ≤ x and σ(a)+σ(b) = σ(a+b). Erdos asked whether S(x) ~ cx. The opposite extreme holds: for every R > 0, S(x)/(x(log x)^R) → ∞, so the count beats every fixed logarithmic scale.

After Chen-He-Ye-Yuan's counterexample to direct three-block ADMM, the subclass in which the third constraint block is the identity matrix remained unresolved: the literature contained neither a convergence proof nor a counterexample.…

Reading computed the order dimension of the poset of regions for most finite Coxeter arrangements, observed that an exceptional type whose dimension exceeds its rank would be the first known simplicial arrangement with that property, and…

Heil, Ramanathan and Topiwala conjectured in 1996 that any finite set of time-frequency shifts of a nonzero square-integrable function is linearly independent. This refutes it: there is a Schwartz function admitting 12 linearly dependent…

Mossel and Peres showed that a single-variable function admits a finite-automata Bernoulli factory exactly when it is rational, and their Theorem 2.9 claimed the same extension to multivariable functions. The multivariable claim is false,…

Along the heat flow, do the successive time derivatives of the entropy of X + √t Z alternate in sign, as conjectured by Cheng and Geng? An explicit measure on R has a fifth derivative with the forbidden sign.

For the ergodic problem tfrac12|Dφ^ε|^2 + F(x) - εΔφ^ε = c(ε) on the torus, normalized by φ^ε(0) = 0, Jauslin, Kreiss and Moser asked whether the vanishing-viscosity limit lim_ε → 0φ^ε always exists. It need not: there is a one-dimensional…

The near-quadratic Elekes-Ronyai expander conjecture over R predicts that a nonspecial polynomial expands any finite set to near-quadratic size. False: a fixed nonspecial quadratic polynomial, together with arbitrarily large finite sets of…

Do the tree-level amplitudes A_n(1^-, 2^+, …, n^+) vanish identically, or can they be nonzero in half-collinear kinematics - and if nonzero, what is their all-n closed form?

Iterates of a firmly nonexpansive operator converge weakly but not strongly, by Genel and Lindenstrauss. Whether their Cesaro means converge strongly was open. They need not: an explicit curve gives a counterexample.

Ross introduced S-perfect numbers, integers expressible as 1 + ∑ λ_j d_j over their proper divisors with coefficients in S, and conjectured that they have the same density as the nondeficient numbers, plus a second conjecture relating odd…

For a simple 3-polytope with at least three faces of size at least 7, must p_6 ≥ 39/20 + p_3/2 - p_5/4 - ∑_k ≥ 7 p_k? Five minimal ten-face counterexamples refute the printed inequality.

The Howland-Kato conjecture that every nonzero positive commutator i[f(P),g(Q)] must arise from functions in appropriate Kato classes is false: i[arctan(P),arctan(Q)] is nonzero and nonnegative.

Is the depth of the mod-p cohomology ring of every finite group realized as the dimension of one of its associated primes? For G = SmallGroup(128, 859) over overlineF_2 the ring has depth 2 while every associated-prime quotient has…