Problems

Among sufficiently large one-separated planar point sets, does the triangular lattice maximize the number of distances below each threshold? Explicit rational oblique lattices beat the triangular lattice under several closed- and…

Does the Dinitz-Garg-Goemans cost-preserving unsplittable-flow rounding conjecture survive on acyclic planar instances with only four terminals? An explicit instance answers no: every cost-nonincreasing unsplittable routing has upper…

For the released four-terminal planar acyclic single-source unsplittable-flow gadget, require one unsplittable routing to be no more expensive than a prescribed fractional routing under each of m strictly positive full-demand…

Does there exist a single real-valued, divergence-free, time-independent Lipschitz velocity field u∈ W^1,∞( T^3; R^3), chosen independently of magnetic diffusivity, that is a fast dynamo for the kinematic induction equation on the flat…

Must every graph with n vertices and δ n^2 edges contain large subgraphs in which every two edges lie on specified short cycles? A dense high-girth construction refutes the statement when δ may shrink with n.

For every connected graph, is the deviation of its adjacency eigenvalues at most its order divided by its average distance? Exact lollipop-graph certificates refute the inequality when deviation means population standard deviation, under…

Let A⊂N be infinite. Must there exist some k≥ 1 such that almost all integers have a divisor of the form a+k for some a∈ A? The question as posed follows negatively from Davenport–Erdős (1951). The AI result settles Tenenbaum's harder…

If n planar points have no four concyclic, must some point determine (1 - o(1))n distinct distances? Failing that, can one always force more than (1/3 + c)n?

The paper proves a closed-string remodeling statement for the affine binary dihedral Calabi–Yau orbifold threefold, a target outside the toric setting of the Bouchard–Klemm–Mariño–Pasquetti remodeling conjecture, replacing the toric mirror…