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.
Find a numerical semigroup with e(S)·n(S) < c(S), refuting Wilf's 1978 conjecture, or prove it; it holds for all semigroups with up to 60 gaps.
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Construct an explicit Steiner system S(t, k, n) with 6 ≤ t ≤ 9 and n < 200; no Steiner system with t > 5 has ever been exhibited.
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Construct, for arbitrarily large n, more than 1.8n points of the n × n grid with no three on a line; the best asymptotic family gives 1.5n.
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Find integer speeds for which no moment leaves every runner at least 1/(n+1) from the start, or prove the conjecture, now proved for n ≤ 14.
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Find an edge-weighted tree on more than six vertices whose pairwise path lengths are exactly 1, 2, …, n(n−1)/2, or prove that none exists.
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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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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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