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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…

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).

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.

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,…

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.

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.

If n_1 < n_2 < … with n_k+1/n_k ≥ c > 1, must ∑_k 1/F_n_k be irrational? The proposed proof closes the range 1 < c < 2 left open by earlier criteria.

If A is a forbidden-divisor set with |A ∩ [1,x]| = o(√x) and B = b_1 < b_2 < … the sifted set, must x^-1 ∑_b_i < x (b_i+1 - b_i)^2 converge to a finite limit?

If h(r) is the maximal finite exact order attainable by an additive basis of order at most r, what is lim_r → ∞ h(r)/r^2? The candidate proof identifies the sharp limit 1/3.

Let R(3;k) be the least n such that every k-colouring of the edges of K_n contains a monochromatic triangle. Determine lim_k→∞ R(3;k)^1/k (a $250 Erdős prize problem). A superexponential lower bound resolves the problem: the limit is…

Estimate the number F(x) of minimal distinct covering systems whose moduli all lie in [1, x]. The candidate proof gives loglog F(x)/log x → 1, i.e. F(x) = exp(x^1+o(1)).

Let L^nf be the Lagrange interpolation polynomials of a continuous f on the Chebyshev nodes. Prove that, for any closed A⊆ [-1,1], there exists a continuous function f such that A is the set of limit points of L^nf(x).

Let G be a simple connected graph on n≥ 5 vertices. If the maximum over all vertices v of ℓ(v) - the independence number of the subgraph induced by the open neighborhood N(v) - is at most 1, must G be well totally dominated? Answered…

If g_3(n) is the largest size of A ⊆ [1,n] with fewer than three representations of every product a_1 a_2, does its conjectured second-order normalized term converge? The candidate proof gives an explicit limit constant.