Problems

For every finite connected simple graph G, is the order of the largest induced tree at least girth(G) - 1 + ecc(G, center(G)), where the last term is the eccentricity of the centre set? Answered affirmatively, with a Lean proof.

Does there exist A=a_1<a_2<…⊂ N which is a minimal basis of order 2 (every large integer is the sum of 2 elements from A, and no proper subset of A has this property) such that lim_k→ ∞a_k/k^2=c for some c≠ 0? A claimed construction gives…

Determine the leading asymptotic of the largest eigenvalue of the N-Majorana quartic SYK Hamiltonian as N → ∞. The preprint proves λ_1/√N → 4∫_0^∞ g_0(t)^4 dt ≈ 0.32504 almost surely, via the limiting free energy at every fixed positive…

If A(x) counts integers satisfying the Sylow divisor condition, determine the constant c in A(x)/x = exp(-(c + o(1)) √log x loglog x). The claimed exact value is c = 1/(2√log 2).

Does there exist an integer polynomial f of degree at least two and a set A ⊆ Z such that every integer has a unique representation n = a + f(k)? A manuscript claims the thirteenth powers admit a tiling complement.

What is the maximum volume of a convex body in R^n whose centroid is its only interior lattice point? Ehrhart conjectured the extremal value in 1964; the sharp maximum is now determined in every dimension.

Let m_1≤…≤ m_k and n be sufficiently large. If T is a tree on n vertices and G is the complete multipartite graph with vertex class sizes m_1,…,m_k, prove that R(T,G)≤ (χ(G)-1)(R(T,K_m_1,m_2)-1)+m_1.

Question 8 of the First Proof experiment (Abouzaid et al.) asks whether a polyhedral Lagrangian surface with exactly four faces meeting at every vertex necessarily admits a Lagrangian smoothing. The research report assembles…

Is every group sofic - does every group admit approximate finite permutation representations? A central open question of geometric group theory since Gromov introduced soficity: soficity implies Gottschalk's surjunctivity conjecture,…

Every finite connected simple graph G satisfies α(G)≥ r(G)+ln(ρ(G)), where α(G) is the independence number, r(G) is the radius, and ρ(G) is the minimum number of pairwise vertex-disjoint paths whose vertices cover V(G).

Among all nonconstant monic polynomials f whose roots lie in [-1, 1], determine inf_f |x ∈ R : |f(x)| < 1|.

Let A=1≤ a_1< a_2<… be a set of integers such that Abackslash B is complete for any finite subset B and not complete for any infinite subset B. If a_n+1/a_n ≥ 1+ε for all n, must lim_n a_n+1/a_n=(1+√5)/2? Under the reading where the ratio…

Does the value of a two-player quantum game decay exponentially under parallel repetition, as Raz's theorem gives for classical games? Yes: an exponential parallel repetition theorem holds for arbitrary finite two-player quantum games.

Are ICC property (T) groups remembered by their von Neumann algebras - if L(Γ) ≅ L(Λ) for such groups, must Γ ≅ Λ? A counterexample refutes Connes' conjecture that these groups are uniquely determined by their group von Neumann algebras.

How large must y(ε, n) be so that every interval (x, x+y) contains at most ε y integers having a divisor in (n, 2n)? The candidate proof gives the sharp fixed-ε order y = Θ_ε(n), uniformly in the translate.

Given a binary matrix M, decide whether its columns can be permuted so that every row contains at most two blocks of 1s and, if it contains two blocks, they are separated by at most one 0. The claimed theorem proves that this (2,1)-Gapped…

For P_n(z) = ∑_k=0^n ε_k z^k with independent uniform signs, does the number R_n of roots in |z| ≤ 1 satisfy R_n/(n/2) → 1 almost surely? The manuscript proves the strong law with R_n = n/2 + O_ω(n^149/150).

Is the closest vector problem NP-hard to approximate within polynomial factors n^c? Yes for some c > 0: hardness of approximation reaches polynomial factors, with consequences for decoding and related lattice problems - a foundational…

For triangular arrays of nodes a_i^n∈[-1,1] let L^nf be the Lagrange interpolation polynomials of a continuous f, with fundamental polynomials p_i^n. Is there a choice of nodes such that for every continuous f there is some x where…

If f(n) is the maximum total side length of n interior-disjoint squares packed in the unit square, is f(k^2 + 1) = k? An exact rational configuration packs 17 squares with total side length greater than 4, refuting the identity at k = 4.