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.
Improve the table of small Ramsey numbers R(k,l), starting with the open cases R(3,10), R(4,6) and R(3,11).
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Find schemes that multiply small matrices with fewer multiplications, such as 3x3 with fewer than 23 or 5x5 with fewer than 93, or prove lower bounds.
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Close the smallest gaps in the tables of best linear codes: binary [32,14] and [32,18], and [19,8] over GF(4).
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Prove that each of six cubic equations z² + y²z + x³ + … = 0 of size at most 24, the smallest open cases, has infinitely many integer solutions.
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Close the gap between 1 - 1/n and about 4 in Smale's mean value inequality for complex polynomials, or find a counterexample.
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Find a closed form for the site percolation threshold of the square lattice, p_c ≈ 0.5927460508; bond percolation has p_c = 1/2.
Decide whether, for every t, some number appears as a binomial coefficient C(n,k) with k ≤ n/2 in exactly t ways.
Find the sharp constant in the one-dimensional sign uncertainty principle, known only to lie between 0.2025 and 0.3216.
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Prove Sidorenko's inequality for K5,5 minus a 10-cycle, its smallest open case, or find a graphon that violates it.
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Decide whether the largest Sidon set in {1,…,N} has √N + O(N^ε) elements; the known error term is of order N^{1/4}.
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Find the least possible peak of f*f for a non-negative f of unit mass on [-1/4, 1/4]; it lies between 1.2802 and 1.50286.
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Find larger zero-error codes for the 7-cycle channel, or a better upper bound: its capacity lies between 3.25883 and 3.3177.
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Decide whether O(log M/ε²) copies of an unknown quantum state suffice, in every dimension, to estimate M given measurement probabilities to accuracy ε.
Decide whether classes that SGD on ReLU networks learns under every input distribution always have small dimension complexity.
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Characterize the spectra that force every bipartite state with those eigenvalues to be separable, and decide whether a PPT test suffices.
Find sustainable ways to keep or raise food production on salt-affected soils through management, tolerant crops and remediation.
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Narrow the ring loading constant, now between 1.119 and 1.3, with harder instances or a sharper proof.
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Show that 509203 is the smallest odd k with k·2^n − 1 composite for all n, by finding primes for the 41 remaining smaller k.
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Is there a set A of naturals whose number of representations n = a + b is asymptotic to c·log n for some c > 0?
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Measure how long tough plant polymers such as root suberin and pollen sporopollenin persist in farm soils, and whether they could store carbon.
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Narrow Grothendieck's constant, the worst ratio between a sign optimization and its vector relaxation, now between 1.7136 and 1.7822.
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Show that a group of rational matrices with finitely many automorphism orbits is virtually soluble; check two proposed proofs.
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Find seven distinct nonzero rationals such that the product of any two, plus one, is a rational square; sextuples are known, septuples are not.
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Explain why some species track a warming climate quickly while others lag or move the wrong way, using traits, landscapes and interactions.
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Find the constant c in R(3,k) ~ c·k²/log k; it is now known to lie between 1/2 and 1.
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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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Work out which weather, snowpack and soil conditions turn rain on snow into an exceptional flood, and how warming shifts the risk.
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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.