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.
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Little-known open problems from Oberwolfach workshops
Problems posed at problem sessions of workshops at the Mathematisches Forschungsinstitut Oberwolfach and recorded in Oberwolfach Reports 2024–2026. They are recent, specific and rarely attacked, which makes them good targets for a first result.
Source: Oberwolfach Reports (EMS Press). Licence: Oberwolfach Reports are CC BY-SA 4.0; problem descriptions here are in our own words.
By field: Combinatorics 12Algebra 6Geometry 5Probability 3Graph theory 3Logic & formalisation 2Analysis 2Number theory 1Complexity 1Algorithms 1
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.
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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?
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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.
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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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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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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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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?
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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?
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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.
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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?
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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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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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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.
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Prove that some small permutation classes are not D-finite
Classes such as Av(1243, 1324, 1432) are believed to have non-D-finite generating functions. An explicit q-series for one of them is known — can it prove non-D-finiteness?
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Radon-type partitions for unions of convex sets
What is the least f(d, s, t) such that any f points in R^d split into A ∪ B with every union of s convex sets covering A meeting every union of t convex sets covering B? Known: O(dst log(st+1)), and f(2, s, s) ≥ s².
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Reducing 3-PIGEON to 2-PIGEON
Finding three pigeons in one hole versus two: is there a randomized black-box reduction from 3-PIGEON to the PPP-complete 2-PIGEON?
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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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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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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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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 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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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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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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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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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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