Short approximate laws on S_n and SL_n(2)
Find the shortest word w(x, y) that equals the identity on 99% of pairs of elements of S_n (or of SL_n(2)). Known bounds for S_n range from about √n to n^{3+o(1)}.
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Open problems
Each problem states how progress is verified and what counts as a contribution. Besides the problems curated here, the catalogue includes open conjectures from Formal Conjectures (with Lean statements), optimization constants and the AlphaEvolve problems. Know one that belongs here? Propose a problem.
All fields Mathematics 970Computer science 30Physics & astronomy 22Chemistry & materials 13Life sciences 6Earth & society 6
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Shortest paths and cycles with prescribed length modulo m
Is there a polynomial-time algorithm for the shortest s–t path (or cycle) whose length is ℓ mod m? Known for ℓ = 0; open in general.
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Small van der Waerden numbers
Determine W(r,k), the least N such that every r-colouring of {1,…,N} contains a monochromatic k-term arithmetic progression. Only seven non-trivial values are known; the open cases W(2,7), W(3,5), W(4,4) and W(5,3) invite better lower-bound colourings and exact computations.
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Smaller covering designs
Improve upper bounds C(v,k,t) for covering designs listed in the La Jolla Covering Repository.
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Smallest sorting networks for 13+ inputs
Find sorting networks with fewer comparators than the best known for n ≥ 13 inputs, or prove optimality.
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Snake-in-the-box — longest induced paths in hypercubes
Find the longest induced path (snake) in the n-dimensional hypercube Q_n. Optimal lengths are known only up to n = 8 (98); for n = 9–13 new records were set in 2026 and further improvements are open.
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Spectral independence of colourings with Δ + 2 colours
Is the uniform distribution on proper q-colourings of a graph of maximum degree Δ O(1)-spectrally independent for every q ≥ Δ + 2? This would give fast sampling.
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Strict monotonicity of growth rates of permutation classes
If a pattern π properly contains ρ, is the growth rate of Av(π) strictly larger than that of Av(ρ)? A first test case: gr(Av(π)) < gr(Av(π⊕1, 1⊕π)) < gr(Av(1⊕π)).
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Structure of graphs with independence number 2
Fox: does every n-vertex graph with no three independent vertices contain K_{n^c, n^c} and a connected matching of size cn? Connected matchings here are tied to Hadwiger's conjecture.
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Tensor rank of 3×3 matrix multiplication
Find a bilinear algorithm multiplying 3×3 matrices with fewer than 23 multiplications, or raise the lower bound.
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Tensor rank of 4×4 matrix multiplication
Find bilinear algorithms that multiply two 4×4 matrices with fewer multiplications. The records are 48 over Q and C (2025) and 47 over GF(2) (2022).
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The Balan–Jiang constant: ℓ1 decompositions of PSD matrices
How much larger than its entrywise ℓ1 norm can the cheapest decomposition Σ‖x_k‖₁² of a PSD matrix be? The worst ratio C_n is between c√n and O(n).
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The degree–diameter problem for graphs
Find the largest graphs with maximum degree d and diameter k. Records for 3 ≤ d ≤ 20 and 2 ≤ k ≤ 10 are tabulated and mostly far below the Moore bound; whether a Moore graph of degree 57 (3250 vertices) exists is a famous open case.
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The dense-matter equation of state from neutron-star observations
Combine public NICER pulse-profile posteriors, gravitational-wave tidal-deformability constraints and pulsar masses into reproducible, well-calibrated constraints on the neutron-star equation of state.
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The origin of fast radio bursts
Determine which source populations and emission mechanisms produce fast radio bursts, and whether repeating and apparently non-repeating bursts share a common origin, using public CHIME/FRB and other catalogues.
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The Ramsey number R(4,6)
Narrow the gap 36 ≤ R(4,6) ≤ 40. A 2-colouring of K_36 with no red K_4 and no blue K_6 would raise the lower bound; lowering the upper bound needs reproducible exhaustive computation.
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The Ramsey number R(5,5)
Narrow the gap between the known lower and upper bounds for R(5,5), currently 43 ≤ R(5,5) ≤ 46.
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The shortest law on the symmetric group
How long must a nontrivial two-letter word w(x, y) be if it equals the identity for every pair of permutations in S_n? The answer lies somewhere between linear and quasi-polynomial in n.
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The sixth Schur number S(6)
Find the largest N such that {1,…,N} can be split into six sum-free sets. After Heule's 2017 SAT proof that S(5) = 160, the best known bound is S(6) ≥ 536, with a large gap to the upper bound.
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The Thomson problem (minimum-energy charges on a sphere)
Find configurations of n unit point charges on the sphere that minimise Coulomb energy. Global optimality is proven only for a few n, so the tasks are to lower best known energies and to prove new cases optimal.
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Thin or arithmetic: symplectic hypergeometric groups
A hypergeometric group ⟨A, B⟩ ⊂ Sp_{2n}(Z) generated by two companion matrices is either of finite index (arithmetic) or thin. Decide the remaining cases, starting with degree 6.
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Three Wilf-inequivalent classes with the same growth rate
Av(132456), Av(124356) and Av(123546) share a growth rate, and A and C even share counting sequences — but A and B do not. How different are the sequences of A and B?
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Thresholds and decoders for quantum error-correcting codes under circuit-level noise
Improve reproducible, circuit-level-noise thresholds and logical error rates for surface codes and quantum LDPC codes through better codes, syndrome circuits and decoders, benchmarked with open simulators such as Stim.
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Unavoidable k-shuffles in long permutations
Does every sufficiently long permutation contain a run of consecutive entries that splits into k copies of the same pattern? Known thresholds: n0(2) = 6, n0(3) = 12.
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Unimodality of radius functions in median graphs
In a median graph of cube-dimension d, is every local minimum of a weighted radius function within distance c·d a global one? And can the weighted center be found in almost linear time?
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Visible-light photocatalysts for overall water splitting
Find a particulate photocatalyst that splits water with high quantum efficiency under visible light, closing the gap between near-perfect UV performance and the low solar-to-hydrogen efficiency of real panels.
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γ-positivity for duals of symmetric edge polytopes
The h*-polynomials of symmetric edge polytopes are not always γ-positive (counterexample, 2026). What about their polar duals, a class of alcoved polytopes?
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A uniform dilation constant for unimodular triangulations
Is there c_d such that c_d·P has a unimodular triangulation for every lattice d-polytope P? Knudsen–Mumford–Waterman give a dilation depending on P.
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A whole-nervous-system model of C. elegans that reproduces behaviour
Build a connectome-constrained model of the C. elegans nervous system that reproduces measured neural activity and behaviour, and validate it against public imaging and connectome data.
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Antarctic marine ice-sheet and ice-cliff instability
Determine whether marine ice-sheet and ice-cliff instabilities can drive rapid Antarctic retreat this century, and how much they widen sea-level projections.
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Approximation ratio and integrality gap for metric TSP
Find better polynomial-time approximation algorithms for the metric Traveling Salesman Problem and prove the conjectured 4/3 integrality gap of the subtour LP. The best known ratio is 3/2 − ε with ε > 10^−36 (Karlin–Klein–Oveis Gharan).
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Computer-assisted proofs of finite-time singularities in 3D Euler
Establish, check and extend rigorous computer-assisted proofs that smooth solutions of the 3D incompressible Euler equations (and related models) develop singularities in finite time.
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Crossing numbers of complete and complete bipartite graphs
Prove Hill's conjecture cr(K_n) = ¼⌊n/2⌋⌊(n−1)/2⌋⌊(n−2)/2⌋⌊(n−3)/2⌋ and Zarankiewicz's conjecture for K_{m,n}. Exact values are known only for small cases (K_n up to n = 14, K_{m,n} for m ≤ 6 and some m = 7, 8 cases).
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Density functional approximations with chemical accuracy
Construct an exchange-correlation approximation that reaches chemical accuracy (~1 kcal/mol) across broad main-group chemistry at semi-local cost, and show it on public benchmarks.
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Do odd perfect numbers exist?
Decide whether an odd perfect number exists. Any such number exceeds 10^1500 and has at least 10 distinct prime factors; progress tightens these constraints.
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Do random generators of Alt(n) give expanders?
Kassabov built bounded generating sets of the alternating groups whose Cayley graphs are expanders. Do two random elements already do this with high probability?
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Electrochemical ammonia synthesis under ambient conditions
Find a catalyst and cell design that reduce N2 to ammonia electrochemically at ambient conditions with verified, contamination-free rates — and separate real signals from false positives.
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Existence of SIC-POVMs (Zauner's conjecture)
Prove that a symmetric informationally complete POVM (d^2 equiangular lines in C^d) exists in every dimension d, and extend the list of dimensions with exact or numerical solutions.
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Explicit rigid matrices (Valiant's rigidity problem)
Construct explicit n×n matrices that stay high-rank even after many entry changes, with parameters strong enough for Valiant's circuit lower bounds. Random matrices are highly rigid, but no explicit matrix is known to meet the required parameters.
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Frankl's union-closed sets conjecture
Every finite union-closed family of sets other than {∅} has an element lying in at least half of its sets. Since Gilmer's 2022 entropy breakthrough the best proven fraction is about 0.38; closing the gap to 1/2 is open.
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Ground-state phase diagram of the doped 2D Hubbard model
Determine reliably, with controlled numerics, where the doped two-dimensional Hubbard model (with and without next-nearest-neighbour hopping t′) is superconducting, striped or otherwise ordered.
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Hadronic vacuum polarisation in the muon g−2
Resolve the disagreement between lattice-QCD and data-driven (e+e− → hadrons) evaluations of the leading hadronic vacuum polarisation contribution to the muon anomalous magnetic moment.
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High-pressure hydride superconductors — prediction and verification
Establish which high-pressure hydride superconductivity claims are robust and how accurately ab initio electron-phonon theory predicts their critical temperatures.
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Legendre's conjecture
Prove that there is always a prime between n^2 and (n+1)^2. For consecutive cubes the analogue is known beyond an explicit (astronomically large) threshold.
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Mechanism of high-temperature superconductivity in the cuprates
Identify the pairing mechanism and the minimal theory that explains superconductivity, the pseudogap and the strange-metal normal state of the copper-oxide superconductors.
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Mutually unbiased bases in dimension 6
Decide whether four (or seven) mutually unbiased bases exist in C^6; only three are known, and a complete set of seven is widely believed not to exist.
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Narrowing equilibrium climate sensitivity
Narrow the uncertainty in equilibrium climate sensitivity — the long-term warming for a doubling of CO2 — using reproducible analyses of public model output and observational records.
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Prebiotic routes to nucleotides and the RNA world
Establish a coherent, geochemically plausible route from simple feedstocks to activated ribonucleotides and self-replicating RNA under one consistent set of early-Earth conditions.
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RNA 3D structure prediction
Predict three-dimensional RNA structures from sequence with accuracy comparable to protein structure prediction, including targets for which no structural template exists.
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Small-scale problems of cold dark matter
Decide whether the cusp–core, too-big-to-fail and rotation-curve diversity problems of ΛCDM on galaxy scales are explained by baryonic physics, by modified dark-matter properties (e.g. self-interactions), or by observational systematics.
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