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.
Beat Lehmer's 1.17628 with an integer polynomial of smaller Mahler measure above 1, or prove the measures stay away from 1.
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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 the least area of a convex shape that covers every planar set of diameter 1, now known to lie between 0.832 and 0.8440936.
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Find the limiting probability that the largest roots of a random polynomial with Gaussian coefficients are all real.
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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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Improve the bounds on kissing numbers in dimensions up to 48, starting with dimension 5, where 40 ≤ τ(5) ≤ 44.
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Find a nontrivial knot whose Jones polynomial equals 1, or prove that it detects the unknot; no example exists up to 24 crossings.
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Lower the proven upper bound 7.10186 on the irrationality measure of π, which is conjectured to equal 2.
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Decide whether the group of interval exchange transformations of the circle is amenable.
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Find 8 or more points in the plane, no three on a line and no four on a circle, with all pairwise distances integers.
Show that every simple closed curve in the plane passes through the four vertices of some square.
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Give an algorithm that, for h between 2^90 and 2^100, quickly finds an imaginary quadratic field with class number exactly h.
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Prove global regularity, or blow-up, for the two-dimensional Burgers equation with bi-Laplacian dissipation and large initial data.
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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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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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Prove that two competing growth processes on the lattice cannot both grow forever when their speeds differ.
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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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Is there a dense set of points in the plane whose pairwise distances are all rational? Expected no, known only conditionally.
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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Show that 4/n is a sum of three unit fractions with distinct denominators for every n > 2, or find a counterexample.
Show that for every infinite set A of reals some set of positive measure contains no scaled and shifted copy of A.
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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 graphs avoiding a fixed induced subgraph H always contain a clique or independent set of size n^{c(H)}.
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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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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