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.
Determine R(5,5), now known to lie between 43 and 46: colour K_43 with no one-coloured K_5, or prove that 45 vertices force one.
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Show that a linear equation that some finite colouring of the integers avoids can be avoided with a number of colours depending only on its length.
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Determine the quantum capacity of qubit Pauli channels; even the depolarizing channel's zero-capacity threshold is only known to lie between 0.064956 and 1/12.
Find a three-dimensional subspace of seven qubits that corrects any single-qubit error, or prove that no ((7,3,3)) code exists.
Prove or refute that minimum output entropy is additive when one of two channels is an arbitrary, possibly non-unital, qubit channel.
Prove or refute that estimating a local Hamiltonian's ground energy to within a constant fraction of its number of terms is QMA-hard.
Prove that stabilizer quantum MDS codes on qudits of dimension q have length at most q²+1, apart from one even-q family, or find a longer one.
Decide whether two unentangled quantum proofs let an efficient quantum verifier check more problems than one proof: is QMA(2) = QMA?
Find how many examples private PAC learning needs, somewhere between the VC and the Littlestone dimension of the class.
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Show that a real number x for which both 2^x and 3^x are integers must itself be an integer.
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Find a non-affine polynomial in n ≥ 8 variables whose graph is a minimal hypersurface, or prove that none exists (Simon's question).
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Show that if A has density α in F_2^n, then 10A contains a coset of a subspace of codimension O(log(1/α)).
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Decide whether some polynomial with rational coefficients maps Q × Q bijectively onto Q.
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Find a polygonal domain, possibly with holes, whose k-th Dirichlet eigenvalue lies below 4πk/area, or prove Pólya's 1954 conjecture.
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Find the maximum number of k-sets and halving lines of n points in the plane; the known bounds are barely superlinear and O(n^4/3).
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Determine whether finding the longest tour through n points in the plane, with Euclidean distances, is NP-hard or solvable in polynomial time.
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Prove that approximating the permanent of most n×n matrices of independent complex Gaussians is #P-hard, as BosonSampling hardness requires.
Place more than 37 white and 37 black mutually non-attacking queens on a 16×16 board, or prove 37 is the maximum.
Prove that for infinitely many primes p, the number p − 2 has an odd number of prime factors counted with multiplicity.
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Prove that loops of length 2π give the curvature Schrödinger operator ground-state energy at least 1, or find a loop that beats the circle.
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Prove that the deficit of maximal flow through oriented bond percolation behaves like sqrt(2(1 − p)) as p tends to 1.
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Prove that the optimal expected cost of building a spanning tree online, with random edge weights, converges, and find the limit.
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Decide whether shared randomness and one bit of communication reproduce all projective measurements on every partially entangled qubit pair.
Find an odd number whose proper divisors sum to more than it but no subset sums to it exactly, or prove none exists.
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Find a finite p-group of odd order with exactly as many automorphisms as elements, or check the proposed example of order 1009^52359.
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Find a finite set of congruences with distinct odd moduli that covers every integer, or prove that none exists.
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Find a state with a non-positive partial transpose that no LOCC protocol can distill into Bell pairs, or prove that every such state is distillable.
Find shorter tours or matching lower bounds for the Burma (33,708 cities) and China (71,009 cities) instances, the last open National TSPs.
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Prove that for Newtonian point masses, the initial conditions whose motion breaks down in finite time form a set of measure zero.
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Show that in two or more dimensions, −Δ + V keeps a.c. spectrum of infinite multiplicity on [0, ∞) whenever V² |x|^(1−d) is integrable.
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