Open problems and research questions, each a git repository: many come from curated collections and credit them. Routes are branches anyone can start or continue; results count once the checks pass.
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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Find a 4-dimensional polytope with n facets whose graph has diameter greater than n − 4, or prove the Hirsch bound in dimension 4.
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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.
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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.
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Prove that any 2^{n−2}+1 points in general position in the plane contain a convex n-gon; open from n = 7 on.
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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.
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Give an explicit ribbon disk for a slice Gompf–Scharlemann–Thompson knot, a leading candidate counterexample to Fox's slice–ribbon conjecture.
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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.
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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)}.
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Construct Nikodym sets over finite fields that leave out as many points as possible, or prove they must fill almost all of the space.
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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.
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Construct explicit graphs on n vertices with no clique or independent set of size c·log n, matching Erdős's random graphs.
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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.
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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.
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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.
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Find n > 24 such that m + τ(m) ≤ n + 2 for every m < n, where τ counts divisors, or prove that none exists.
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Find a polynomial of degree n with all n+1 coefficients nonzero integers whose square has fewer than √n nonzero coefficients.
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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.
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Find a Costas array of order 32, the smallest order for which none is known, or prove that none exists.
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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.
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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.
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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.
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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.
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