Imaginary quadratic fields whose class groups have large ℓ-rank
Exhibit imaginary quadratic fields whose class groups have record ℓ-rank, such as 3-rank 9 or 5-rank 6, certified by independent forms.
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Exhibit imaginary quadratic fields whose class groups have record ℓ-rank, such as 3-rank 9 or 5-rank 6, certified by independent forms.
Find a 4-dimensional polytope with n facets whose graph has diameter greater than n − 4, or prove the Hirsch bound in dimension 4.
Place n points in a square or triangle so the smallest triangle they form is as large as possible; beat a record or prove one optimal.
Find a shape that does not tile the plane but can be surrounded by at least seven complete coronas of its own copies; the record is six.
Prove that any 2^{n−2}+1 points in general position in the plane contain a convex n-gon; open from n = 7 on.
Find positive integers x, y with 0 < |y² − x³| < √x/100, beating the record ratio of about 46.6 in Hall's conjecture, unbroken since 1998.
Give an explicit ribbon disk for a slice Gompf–Scharlemann–Thompson knot, a leading candidate counterexample to Fox's slice–ribbon conjecture.
Show that every tree with m edges can be labelled by distinct numbers 0 to m so that the edge differences are exactly 1 to m.
Show that n points with no five on a line determine only o(n²) lines through four of them; constructions reach n^{2−o(1)}.
Construct Nikodym sets over finite fields that leave out as many points as possible, or prove they must fill almost all of the space.
Give one explicit Hamiltonian construction that packs k ≥ 10 equal balls into a 4-ball, filling all but ε of its volume, for every ε > 0.
Construct explicit graphs on n vertices with no clique or independent set of size c·log n, matching Erdős's random graphs.
Find k−1 positive k-th powers summing to a k-th power for some k ≥ 6, extending the counterexamples to Euler's conjecture for k = 4 and 5.
Is there a dense set of points in the plane whose pairwise distances are all rational? Expected no, known only conditionally.
Show that 4/n is a sum of three unit fractions with distinct denominators for every n > 2, or find a counterexample.
Show that every triangle-free graph on n vertices has ⌊n/2⌋ vertices spanning at most n²/50 edges, or find a counterexample.
Determine the largest r-uniform hypergraph on n vertices with no k disjoint edges; Erdős conjectured one of two simple families wins.
Show that every finite graph with minimum degree 3 has a cycle of length 4, 8, 16 or another power of two, or find a counterexample.
Find n > 24 such that m + τ(m) ≤ n + 2 for every m < n, where τ counts divisors, or prove that none exists.
Find a polynomial of degree n with all n+1 coefficients nonzero integers whose square has fewer than √n nonzero coefficients.
Construct an infinite Sidon set with at least N^{1/2−ε} elements up to N for every ε > 0, or show that none exists.
Find a Costas array of order 32, the smallest order for which none is known, or prove that none exists.
Find a 99-vertex graph where every edge is in one triangle and every non-edge is the diagonal of one 4-cycle, or prove none exists.
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Show that some vertex of every convex n-gon has at least ⌊n/2⌋ distinct distances to the other vertices, or find a counterexample.
Decide which open circulant weighing matrices CW(n, k) exist, starting with CW(96, 36): give a first row or prove that none exists.
Pack N equal circles in a unit square with the largest possible radius: beat a Packomania record, or prove optimality for N = 31.
Decide whether 6 or 7 colours are needed to colour the plane so that points at distance 1 always get different colours.
Show that every synchronizing automaton with n states has a reset word of length at most (n − 1)².
Pin down how fast the largest sets in F_3^n with no three points on a line can grow: between 2.2203^n and 2.756^n.
Show that a subgraph of the n-cube with no 4-cycle has at most about half of the cube's edges; the best bound is 0.60318.