The problems from Georgiev, Gómez-Serrano, Tao and Wagner, Mathematical exploration and discovery at scale (2025), where AlphaEvolve searched for constructions in analysis, combinatorics and geometry. Each states the best known value; a better construction is verified by recomputing its score.
Source: AlphaEvolve repository of problems. Licence: Text CC BY 4.0, code Apache 2.0.
For any dimension n, let C(n) denote the quantity C(n) := π^n/2/Γ(n/2+ 1) inf_f (r/2)^n f(0)/hat f(0) where f ranges over integrable continuous functions f := ℝ^n → ℝ, not identically zero, with hat f(ξ) ≥ 0 for all ξ and f(x) ≤ 0 for all |x| ≥ r for some r>0.
Let C be the smallest constant such that min_0 ≤ t ≤ 1 ∫_ℝ f(x) f(x+t) dx ≤ C ‖f‖_L^1(ℝ)^2 for f ∈ L^1(ℝ). What is C?
Let C be the best constant for which one has max_-1/2 ≤ t ≤ 1/2|∫_ℝ f(t-x) f(x) dx| ≥ C (∫_-1/4^1/4 f(x) dx)^2 for all f : [-1/4,1/4] → ℝ (note f can take negative values). What is C?
Let n ≥ 1. Let C(n) be the largest displacement that the n^th block in a stack of identical rigid rectangular blocks of width 1 can be displaced horizontally over the edge of a table, with the stack remaining stable.
For any 1 ≤ p < ∞ and n ≥ 2, let C(p,n) be the smallest constant such that for any complex polynomial f of degree n with zeroes z_1,…,z_n satisfying 1/n Σ_i=1^n |z_i|^p ≤ 1, and every zero f(ζ)=0 of f, there exists a critical point f'(ξ) = 0 of f with |ξ - ζ| ≤ C(p,n). What is C(p,n)?
For n ≥ 4, let Ω(n) be the set of pairs (α,β) ∈ ℝ_+^2 such that, whenever P is a degree n polynomial whose roots z_1,…,z_n sum to zero, and ξ_1,…,ξ_n-1 are the critical points (roots of P'), that |ξ_1|^4 + … + |ξ_n-1|^4 ≤ α (|z_1|^4 + … + |z_n|^4) + β (|z_1|^2 + … + |z_n|^2)^2. What is Ω(n)?
For any natural number n, let Δ(n) be the size of the smallest set B of integers such that every natural number from 1 to n is expressible as a difference of two elements of B (such sets are known as difference bases for the interval 1,…,n). Write C(n) := Δ^2(n)/n, and C := inf_n ≥ 1 C(n).
Is it true that every convex polygon has a vertex with no other 4 vertices equidistant from it?
For any natural number N, let C(N) denote the largest cardinality of a subset A of 1,…,N with the property that ab+1 is square-free for all a,b ∈ A. Establish upper and lower bounds for C(N) that are as strong as possible.
For any natural n, let C(n) denote the maximum possible sum of side lengths of n squares with disjoint interiors contained inside a unit square. Obtain upper and lower bounds for C(n) that are as strong as possible.
Let G be a finite graph with minimum degree at least 3. Must G contain a cycle of length 2^k for some k ≥ 2?
For a natural number N, let C(N) be the largest quantity such that N! can be factored into N factors that are greater than or equal to C(N) (see OEIS A034258). Establish upper and lower bounds on C(N) that are as strong as possible.
Let 1 ≤ q ≤ ∞, and let j and m be non-negative integers such that j < m. Furthermore, let 1 ≤ r ≤ ∞, p ≥ 1 be real and θ ∈ [0, 1] such that the following relations hold: 1/p = j + θ ( 1/r - m ) + 1 - θ/q, j/m ≤ θ < 1.
For n ≥ 1, let U_n denote the set of polynomials p(z) of degree n with coefficients ± 1.
If n,m are natural numbers, let C(n,m) denote the maximum number of incidences that are possible between n points and m lines in the plane. Establish upper and lower bounds on C(n,m) that are as strong as possible.
Let C denote the best constant for which | x: sup_h>0 1/2h ∫_x-h^x+h f(y) dy ≥ λ | ≤ C/λ ∫_ℝ f(x) dx for absolutely integrable non-negative f : ℝ → ℝ. What is C?
For 1 ≤ p ≤ 2, let C(p) be the best constant such that ‖ hat f ‖_L^p'(ℝ) ≤ C(p) ‖ f ‖_L^p(ℝ) holds for all test functions f : ℝ → ℝ. Here p' := p/p-1 is the dual exponent of p. What is C(p)?
For any n ≥ 3 and any convex body K in the plane, let C(n,K) be the largest quantity such that in every configuration of n points in K, there exists a triple of points determining a triangle of area at most C(n,K) times the area of K. Establish upper and lower bounds on C(n,K).
For any n ≥ 3 let C(n) be the largest quantity such that in every configuration of n points in the plane, there exists a triple of points determining a triangle of area at most C(n) times the area of their convex hull. Establish upper and lower bounds on C(n).
Let C be the best constant for which one has ‖f f‖_L^2(ℝ)^2 ≤ C ‖ff‖_L^1(ℝ) ‖f * f‖_L^∞(ℝ) for non-negative f : ℝ → ℝ. What is C?
Let d ≥ 1, and let q be a prime power. Let 𝔽_q be a finite field of order q. A Kakeya set is a set K that contains a line in every direction, and an Nikodym set N is a set with the property that every point x in 𝔽_q^d is contained in a line that is contained in N ∪ x.
Let n ≥ 2. Let C^T(n) denote the minimal area |bigcup_j=1^n T_j| of a union of triangles T_j with vertices (x_j,0), (x_j + 1/n, 0), (x_j + j/n, 1) for some real numbers x_1,…,x_n, and similarly define C^P(n) denote the minimal area |bigcup_j=1^n P_j| of a union of parallelograms P_j with vertices…
For positive-semidefinite d × d matrices A_1, …, A_n and any unitarily invariant norm |||·||| (including the operator norm and Schatten p-norms) and m ≤ n, define C(n,m,d) := inf frac 1/n^m Σ_j_1, j_2, …, j_m = 1^n |||A_j_1A_j_2… A_j_m||| (n-m)!/n! Σ_substackj_1, j_2, …, j_m = 1 \ all distinct^n…
Let n,d ≥ 2. Let C(d,n) denote the largest quantity such that, given any n distinct points x_1,…,x_n in R^d, the maximum distance max_1 ≤ i < j ≤ n ‖x_i-x_j‖ between the points is at least C(d,n) times the minimum distance min_1 ≤ i < j ≤ n ‖x_i-x_j‖. Establish upper and lower bounds for C(d,n).
For 0 ≤ ρ ≤ 1, let C(ρ) denote the largest quantity such that any graph on n vertices and (ρ+o(1)) C(n, 2) edges will have at least (C(ρ)-o(1)) C(n, 3) triangles. What is C(ρ)?
For any n ≥ 1 and a geometric shape P (e.g. a polygon, a polytope or a sphere), let C(n, P) denote the smallest scale s such that one can place n identical copies of P with disjoint interiors inside another copy of P scaled up by a factor of s.
Is it possible for seven infinite circular cylinders C_1,…,C_7 of unit radius to touch all the others?
For any n ≥ 4, Let C(n) denote the maximum volume of a polyhedron with n vertices that all lie on the unit sphere S^2. What is C(n)? Which polyhedra attain the maximum volume?
Let d ≥ 2 and D ≥ 1. For p ∈ 4,∞, let C^p(d,D) be the maximum of the ratio frac‖u‖_L^p(S^d)‖u‖_L^2(S^d) where u ranges over (real) spherical harmonics of degree D on the d-dimensional sphere S^d, which we normalize to have unit measure.
For each n ≥ 2, let C(n) be the smallest constant such that for any complex polynomial f of degree n ≥ 2 with zeros z_1, …, z_n in the unit disk and critical points w_1, …, w_n-1, and for any nonnegative weights l_1, …, l_n ≥ 0 satisfying Σ_k=1^n l_k = 1, we have min_1 ≤ j ≤ n-1 | Σ_k=1^n l_k z_k -…
A graphon is a symmetric measurable function W : [0,1]^2 → [0,1]. Given a graphon W and a finite graph H = (V(H),E(H)), the homomorphism density t(H,W) is defined as t(H,W) = ∫_[0,1]^V(H) Π_v,w ∈ E(H) W(x_v,x_w) Π_v ∈ V(H) dx_v.
For n ≥ 2, let C(n) be the least constant such that for any polynomial f of degree n, and any z ∈ ℂ with f'(z) ≠ 0, there exists a critical point f'(ξ)=0 such that |f(z)-f(ξ)/z-ξ| ≤ C(n) |f'(z)|. Establish upper and lower bounds for C(n) that are as strong as possible.
A spherical t-design on the d-dimensional sphere S^d ⊂ R^d+1 is a finite set of points X ⊂ S^d such that for any polynomial P of degree at most t, the average value of P over X is equal to the average value of P over the entire sphere S^d.
For n a natural number, let C(n) denote the size of the largest subset of [n]^2 = 1,…,n^2 that does not contain a (possibly flat) isosceles triangle. In other words, C(n) := max_S⊂ [n]^2|S|: a,b,c∈ S distinct implies ‖a-b‖ ≠ ‖b-c‖.
Given a natural number N and a ring R of size at least N, let C(R, N) denote the least possible value of max(|A+A|, |A · A|) where A ranges over subsets of R of cardinality N. Establish upper and lower bounds for C(R, N) that are as strong as possible.
For N ≥ 2, let C(N) denote the maximal value of the energy E(z_1,…,z_N) := min_1 ≤ i < j ≤ N ‖z_i-z_j‖ where z_1,…,z_N range over points in S^2. Establish upper and lower bounds on C(N) that are as strong as possible. What type of configurations z_1,…,z_N come close to achieving the maximal energy?
For each slope r ∈ ℝ ∪ ∞ define the projection π_r : ℝ^2 → ℝ by π_r(a,b) = a + rb for r ≠ ∞ and π_∞(a,b)=b.
Let C be the largest quantity such that, as n → ∞, one can locate a 3-uniform hypergraph on n vertices and at least (C-o(1)) C(n, 3) edges that contains no copy of the tetrahedron K^(3)_4. What is C?
For n a natural number, let C(n) denote the size of the largest subset of [n]^3 = 1,…,n^3 such that no 5 points lie on a sphere or a plane. Obtain upper and lower bounds for C(n) that are as strong as possible.
Let C denote the infimal value of λ_0(γ), the least eigenvalue of the Schrödinger operator H_γ = -d^2/ds^2 + κ^2(s) associated with a simple closed convex curve γ parameterized by arclength and normalized to have length 2π, where κ(s) is the curvature.
Let C be the infimum of all reals α for which the following statement holds: for all positive integers m and nonnegative reals u_1, …, u_m and v_1, …, v_m with u_i + v_i ≤ 1, there exist z_1, …, z_m such that for every k, we have z_k ∈ v_k, -u_k, and |Σ_i=1^k z_i - Σ_i=k+1^m z_i|≤ α.
Define C to be the largest volume of a connected bounded subset S_3 of R^3 that can continuously pass through a three-dimensional snake-shaped corridor with a unit square cross-section, consisting of two turns in the x-y and y-z planes that are far apart. What is C?
Given a function f ∈ L^1(ℝ), set A(f) := inf r > 0: f(x) ≥ 0 hbox for all |x| ≥ r . Let C be the largest constant for which one has A(f) A(hat f) ≥ C for all even f with f(0), hat f(0) < 0. Establish upper and lower bounds for C that are as strong as possible.
Let 1 ≤ p,q,r ≤ ∞ with 1/r + 1 = 1/p + 1/q. Let C(p,q,r) denote the supremum of the quantity Q(f, g) := ‖f * g‖_r/‖f‖_p ‖g‖_q over all non-zero test functions f,g. What is C(p,q,r)?
