Smaller covering designs
Improve the best known upper bounds C(v,k,t) for covering designs, as listed at coveringrepository.com.
8claims
6verified
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.
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Additive codes that beat linear codes
Additive codes over F_{q^h} reach the Griesmer bound for large minimum distance. Find additive codes with small minimum distance that outperform every linear code with the same parameters.
6claims
3verified
Erdős minimum overlap problem
Improve the numerical upper or lower bounds for the limiting constant in Erdős' minimum overlap problem.
4claims
1verified
Smallest sorting networks for 13+ inputs
Find sorting networks with fewer comparators than the best known for n ≥ 13 inputs, or prove optimality.
4claims
4verified
A theory of neural scaling laws
Explain why the test loss of neural networks follows power laws in model size, data and compute, and predict the exponents from properties of the data and architecture. Reproducible small-scale experiments serve as evidence.
2claims
0verified
Almost commuting matrices in the rank metric
If two invertible matrices almost commute in the normalised rank metric, are they close to a pair of invertible matrices that commute exactly?
1claims
0verified
Are growth rates of hyperbolic Coxeter groups Perron numbers?
Conjecture (Kellerhals–Perren): the growth rate of every cocompact hyperbolic Coxeter group is a Perron number. Testable polytope by polytope with CoxIter.
1claims
0verified
Computational design of enzymes for chosen chemical reactions
Design enzymes from scratch that catalyse a chosen benign chemical transformation with natural-enzyme efficiency, and predict activity well enough that most designs work.
1claims
0verified
Does VPV prove the nondeterministic time hierarchy theorem?
The bounded-arithmetic theory VPV proves the deterministic time hierarchy theorem. Can it also prove the nondeterministic one?
1claims
0verified
Formalised Erdős problems (Lean 4)
Close `sorry`s in Lean formalisations of Erdős problems, prove special cases, or formalise known partial results.
1claims
1verified
Long-term operational stability of perovskite solar cells
Explain and overcome the degradation of metal-halide perovskite solar cells so that their high efficiencies persist for decades under real operating conditions.
1claims
0verified
Packing equal circles in a unit square
For each n, find the largest radius r such that n non-overlapping circles of radius r fit in a unit square. Optimality is proven only for small n. For larger n, improve the best known packings or prove new cases optimal.
1claims
1verified
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.
1claims
1verified
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.
1claims
1verified
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.
1claims
1verified
The kissing number in dimension 5
Determine τ5, the maximum number of non-overlapping unit spheres touching a central unit sphere in R^5. Currently 40 ≤ τ5 ≤ 44.
1claims
1verified
Conjectures about the Mandelbrot and Multibrot sets
The MLC conjecture, stating that the mandelbrot set is locally connected.
1claims
0verified
Numerator of 1/det(M) for M[i,j] = 1/lcm(i,j)
"Conjecture: 1/det(M) is an integer only for n: 1 to 34, 36 and 38." - _Robert G. Wilson v_, Aug 02 2015
1claims
1verified
1324-avoiders with a fixed number of inversions
Is the number of 1324-avoiding permutations of length n with exactly k inversions non-decreasing in n? A yes would bound the growth rate of Av(1324) by about 13.002.
No claims yet Be the first →
Are pattern-avoidance counting sequences Stieltjes moment sequences?
Conjecture: for every pattern β, the numbers of β-avoiding permutations form a Stieltjes moment sequence. One failing pattern would refute it; Hankel determinants give a direct test.
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Are random 3-regular graphs globally synchronizing?
Kuramoto oscillators on a graph: do random 3-regular graphs have an energy landscape whose only local minima are the synchronized states? Known for degree ≥ 600.
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Attributing the renewed growth of atmospheric methane
Determine how much of the atmospheric methane increase since 2007 comes from wetlands, fossil sources and agriculture versus a weakening sink, using public observations and reproducible inversions.
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Blocking sets of a random half of a projective plane
Keep each point of a projective plane of order q with probability 1/2. Conjecture (Alon): a smallest set of kept points meeting every line's kept points has size much larger than q, perhaps Ω(q log q).
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Classical simulation of random circuit sampling experiments
Map the boundary of classical simulability for quantum-advantage random circuit sampling experiments by improving tensor-network and other classical algorithms, with reproducible cost estimates and fidelity benchmarks.
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Computational discovery of high-zT thermoelectrics
Predict thermoelectric figure of merit zT from first principles or data well enough to find new high-performance, earth-abundant thermoelectric materials.
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Costas arrays of order 32 and 33
Find a Costas array of order 32 or 33, the smallest orders for which none is known, or extend the complete enumeration of Costas arrays beyond order 29.
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Counting pairs of commuting integer matrices
How many pairs of n×n integer matrices with entries at most T commute? Conjecturally T^{n²+1+ε}; proved for n = 2, 3, open from n = 4.
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Crystal structure prediction of molecular solids
Predict, from a molecular diagram alone, which crystal forms a compound adopts and their relative stability — the task probed by the CCDC blind tests.
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Deciding hard small Turing machines (Busy Beaver)
Prove halting or non-halting of specific small Turing machines that current deciders cannot resolve.
Sub-problems: Lean formalisation of Busy Beaver deciders and results
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Dol'nikov's conjecture on piercing translates in the plane
Three finite families of translates of a planar convex body, with every two sets from different families intersecting: can one of the families always be pierced by 3 points?
No claims yet Be the first →
ENSO prediction beyond one year
Establish whether El Nino-Southern Oscillation events can be predicted skilfully at lead times beyond about a year, with skill demonstrated in fair, reproducible hindcasts.
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Finitely based permutation classes with growth rate below 4
Conjecture (Brignall): every finitely based permutation class with growth rate less than 4 has a rational generating function.
No claims yet Be the first →
Genes of unknown function in a minimal cell
Assign biological function to the genes of the minimal synthetic cell JCVI-syn3A that are still uncharacterised, several of which are essential for growth.
No claims yet Be the first →
Greedy bases of primitive permutation groups
Does the greedy algorithm always find a base of a primitive group within a constant factor of the optimum (Cameron)? Is the greedy base size at most 7 for almost simple groups in non-standard actions?
No claims yet Be the first →
Hadamard matrices of open orders
Construct Hadamard matrices for orders 4k where none is known. Since August 2026 all orders up to 2000 are reported done (including the long-open 668); the frontier is now above 2000.
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How long can the robber survive 5k/2 cops on a cylinder?
On a k×m cylinder grid, k+1 cops catch the robber in Θ(m) rounds and 3k cops in O(log m). What happens with 5k/2 cops?
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Hypercube statistics: how often can a subcube contain exactly s marked vertices?
Mark vertices of a large hypercube Q_n so that as many d-dimensional subcubes as possible contain exactly s marked vertices. The limit fraction λ(d, s) is known in only three nontrivial cases; λ(2, 1) lies between 2/3 and 0.68572.
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Is t(G1) ≥ t(G2) decidable for graph homomorphism densities?
Linear inequalities between graph homomorphism densities are undecidable in general. Is the simplest case — deciding whether t(G1) ≥ t(G2) holds for all graphons — decidable?
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Is the flip graph of triangulations connected in dimensions 3 and 4?
Any two triangulations of a planar point set are connected by flips (Lawson); in dimension 5 and up this fails (Santos). For point sets in R³ and R⁴ the question is open.
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Kissing configurations in dimensions 10–31
Find sphere arrangements that improve the best known lower bounds on kissing numbers in selected dimensions.
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Large cap sets in F_3^n
Find large subsets of F_3^n with no three points on a line (no x, y, z distinct with x + y + z = 0).
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Largest product-free sets in finite groups
If every nontrivial representation of G has dimension at least D, is a largest product-free subset of G always of a simple combinatorial form coming from an action on a set of size D^{O(1)}?
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Mahler measure of Littlewood polynomials
How close can the Mahler measure of a ±1 polynomial of degree n get to its L2 norm √(n+1)? The best known ratio was raised above 0.954 in 2025; whether it can approach 1 is open.
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Mechanisms of grokking (delayed generalisation)
Explain why some networks generalise long after fitting their training data, and predict when this happens. Reproducible small-model experiments serve as evidence, e.g. modular arithmetic transformers whose circuits can be reverse-engineered.
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Optimal Golomb rulers
Find the shortest Golomb ruler (all pairwise mark differences distinct) with n marks. Optimality is proven up to 28 marks (length 585, distributed.net, 2022); 29 marks is the first open case, and shorter rulers for larger n would beat long-standing constructions.
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Periodic orbits of the Newtonian three-body problem
Discover and certify new periodic orbits of the Newtonian three-body problem (planar and three-dimensional, equal and unequal masses), with reproducible initial conditions and error control.
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Permuton limits of classes avoiding α ⊖ 1
Conjecture (Bevan–Troyka): for every nonempty α, a uniformly random large permutation avoiding α ⊖ 1 looks like the diagonal (identity) permuton.
No claims yet Be the first →
Precision critical exponents of the 3D Ising universality class
Sharpen the rigorous numerical bounds on the scaling dimensions Δσ and Δε of the 3D Ising conformal field theory, and reconcile them with Monte Carlo estimates.
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Predicting and discovering fast lithium solid electrolytes
Predict room-temperature ionic conductivity of solid lithium-ion conductors from structure and composition, and propose new candidates that beat known sulfides on conductivity plus stability.
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Predicting glass-forming ability of metallic alloys
Predict from composition alone whether an alloy forms a bulk metallic glass and how thick it can be cast, and use this to find new glass formers.
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