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 function N → Z with bounded steps whose graph has no three-term progression, or prove none exists (the additive square problem).
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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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Prove that the self-describing Kolakoski sequence of 1s and 2s has a density of 1s, equal to 1/2.
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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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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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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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Show that if every large integer is a sum of two elements of A, some integers have arbitrarily many such representations.
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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 graphs avoiding a fixed induced subgraph H always contain a clique or independent set of size n^{c(H)}.
Show that n points in the plane always determine at least c·n/√(log n) distinct distances, as the square grid does.
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Find lim R(k)^{1/k} for the diagonal Ramsey numbers, or even prove that the limit exists; it lies between √2 and 3.8.
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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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Count the polyominoes with n cells, known exactly up to n = 70, and pin down Klarner's growth constant, now between 4.0025 and 4.5252.
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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.
Show that if A, B ⊆ {1,…,N} each have at least N^0.49 elements, then the sumset A + B contains a composite number.
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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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Pin down the constant γ in the expected longest common subsequence γn of two random binary strings, rigorously between 0.79266 and 0.82628.
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Show that every synchronizing automaton with n states has a reset word of length at most (n − 1)².
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Show that an open subset of the unit square with area α contains the corners of an axis-parallel rectangle of area at least cα².
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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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Decide whether every bipartite graph G has ex(n;G) ~ c·n^α for some α in [1,2), and whether α must be rational.
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Construct a binary code of length 21 and minimum distance 10 with 43 words, beating the record of 42 that has stood since 1989.