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.
159 shown· page 3 of 4
The moving sofa constant C_41a=A is the maximum area of a connected, rigid planar shape that can maneuver through an L-shaped corridor of unit width. The corridor is formed by two semi-infinite strips of width 1 meeting at a right angle.
For integers m,n≥ 1, let B_ℝ,m(n) be the smallest constant such that every m-linear form T:(ℓ_∞^n)^m → ℝ satisfies the (multilinear) Bohnenblust–Hille inequality (Σ_j_1,…,j_m=1^n bigl|T(e_j_1,…,e_j_m)bigr|^2m/m+1)^m+1/2m ≤ B_ℝ,m(n) ‖T‖, where ‖T‖:=sup_‖x^(1)‖_∞,…,‖x^(m)‖_∞ ≤…
Let X be an integrable real random variable. We say that X is 1-sub-Gaussian in the tail sense if E[X]=0 quadand ℙ(lvert Xrvert>t)≤ 2e^-t^2/2quadfor all t≥ 0.
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 χ be a primitive Dirichlet character modulo q, and define S(χ) := max_N≤ q lvertΣ_1≤ n≤ Nχ(n)rvert. The Polya-Vinogradov inequality states that S(χ) ≤ c √(q) log q for some absolute constant c. <a href="#BK2020-def-PV">[BK2020-def-PV]</a> For squarefree moduli, define C_72^even (resp.
Given a real matrix A, let its condition number be κ(A):=σ_max(A)/σ_min(A), where σ_min(A) and σ_max(A) denote the smallest and largest singular values of A, respectively (with κ(A)=∞ if σ_min(A)=0).
C_46 is the infimal exponent p such that one has the global bound ‖widehatf dσ‖_L^p(ℝ^3) lesssim_p ‖f‖_L^∞(S^2) qquadfor all f∈ L^∞(S^2).
For subsets K,L⊂ℝ^n, their Minkowski sum is K+L := x+y: x∈ K, y∈ L. In general, one cannot expect a reverse Brunn-Minkowski inequality for arbitrary compact sets, even with a fixed multiplicative constant.
C_45 is the asymptotic density (if it exists) of the set of odd integers that can be expressed as the sum of a prime number and a power of two.
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 -…
An algebraic integer α of degree d, with conjugates α_1,…,α_d, is totally positive if all of its conjugates are real and strictly positive. Its absolute trace (or trace-to-degree ratio) is overlinetr(α) := tr(α)/deg(α) = 1/dΣ_i=1^d α_i . Let A denote the set of totally positive algebraic integers.
Let Γ⊂ SL_2(ℤ) be a congruence subgroup. Denote by 0=λ_0<λ_1(Γ)≤ λ_2(Γ)≤ ⋯ the eigenvalues of the (non-Euclidean) Laplacian acting on L^2(ΓbackslashH).
Let f:ℂ→ℂ be a polynomial of degree n≥ 2 whose zeroes all lie in the closed unit disk D(0,1)=\z:lvert zrvert≤ 1\. Sendov's conjecture states that if λ_0 is one of these zeroes, then f' has at least one zero in D(λ_0,1). every zero λ_0 of f has a critical point in D(λ_0,1).
Let C_7 denote the cycle graph on 7 vertices. We define C_9 to be the Shannon capacity of mathcal C_7: C_9 := Θ(mathcal C_7), where for a graph G, the Shannon capacity Θ(G) is defined by Θ(G) := sup_n ≥ 1 α(G^boxtimes n)^1/n.
C_5b is the largest constant such that every (4,5)-set of size n (i.e., a set of reals such that every four-element subset determines at least five distinct differences) contains a Sidon set of cardinality C_5bn.
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 a finite nonempty subset A of an abelian group, write σ(A) := |A+A|/|A|, δ(A) := |A-A|/|A| for the doubling and difference constants. Ruzsa [Ru96] proved δ ≤ σ^2, and the Plünnecke–Ruzsa inequalities give the converse σ ≤ δ^2 [Bl26].
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.
For a bounded set X⊂ ℝ^n, its diameter is diam(X) := sup‖x-y‖_2: x,y∈ X. Let b(X) be the smallest integer m such that X can be written as a union X = X_1 ∪ ⋯ ∪ X_m with diam(X_i) < diam(X) for all i=1,…,m.
C_10c is the least constant K for which one has disc(A) ≤ K√(n) qquadfor all n and all A∈[-1,1]^n× n. Here the discrepancy disc(A) is defined as disc(A) := min_x∈± 1^n ‖Ax‖_∞.
C_36=Δ_4 is the (optimal) sphere packing density in ℝ^4, i.e. the largest fraction of ℝ^4 that can be covered by congruent balls with disjoint interiors.
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.
Let ℤ^2 denote the square lattice graph with vertex set ℤ^2 and edges between nearest neighbors (Euclidean distance 1). A self-avoiding walk (SAW) on a graph G=(V,E) is a walk that visits no vertex more than once.
Let Av_n(1324) be the set of permutations of \1,2,…,n\ that avoid the permutation pattern 1324, and let S_n(1324) := |Av_n(1324)|. <a href="#CJS12-def-Sn">[CJS12-def-Sn]</a> The Stanley–Wilf limit (growth constant) for the pattern 1324 is C_30 := lim_n→∞ bigl(S_n(1324)bigr)^1/n.
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‖.
For a finite set A ⊂ ℝ write A+A = \ a+b : a,b ∈ A \, AA = \ ab : a,b ∈ A \ for the sumset and product set. The (real) sum-product exponent is C_84b := liminf_n → ∞ min_substackA ⊂ ℝ \ lvert Arvert = n log max(lvert A+Arvert, lvert AArvert)/log n.
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.
C_12 = β_2 is the constant such that the length L_n of the shortest tour through n independent uniform random points satisfies L_n/√(n)→ β_2 almost surely.
Let X_1,X_2,… be i.i.d. real random variables with E X_1 = 0, Var(X_1)=1, and finite third absolute moment β_3 := E|X_1|^3 < ∞. Let S_n := X_1+⋯+X_n/sqrt n, F_n(x):=ℙ(S_n≤ x), and let Φ denote the standard normal distribution function.
Let G be a simple graph. The acyclic chromatic index χ_a'(G) of G is defined to be the least number of colors needed to color the edges of G so that no two edges coincident on the same vertex are homochromatic and there is no cycle whose edges are colored with only two colors.
The complex Grothendieck constant (often denoted K_G^ℂ) is the smallest number C_10b such that, for every m,n≥ 1 and every complex matrix A=(a_ij)∈ℂ^m× n, max_substacku_1,…,u_m∈ S^∞\ v_1,…,v_n∈ S^∞ |Σ_i=1^mΣ_j=1^n a_ij⟨ u_i, v_j⟩| ≤ C_10b\ max_substack|s_1|=⋯=|s_m|=1\ |t_1|=⋯=|t_n|=1…
Let Q_n = -1,1^n be the Hamming cube (two vertices are adjacent if they differ in exactly one coordinate). For a set A ⊂ Q_n define the function h_A:Q_n→ 0,1,...,n by - h_A(x)=0 if x∉ A; - if x∈ A, then h_A(x) is the number of neighbors of x that lie in the complement A^c.
C_2 is the Crouzeix constant (sometimes denoted Q). It is the smallest constant C such that for every n ≥ 1, every complex matrix A ∈ ℂ^n × n, and every complex polynomial p one has ‖p(A)‖ ≤ C max_z ∈ W(A) |p(z)|, where ‖·‖ is the operator norm induced by the Euclidean norm (i.e.
C_3a is the largest constant such that there exist arbitrarily large sets A,B of integers such that |A+B| ≪ |A| and |A-B| ≫ |A+B|^C_3a.
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 a real number γ, its irrationality exponent μ(γ) is defined by μ(γ) := infBiglc∈ℝ: Bigllvertγ-a/bBigrrvert≤ lvert brvert^-c has only finitely many solutions (a,b)∈ℤ^2Bigr. <a href="#Zud2004-def-mu">[Zud2004-def-mu]</a> We define C_7b := μbigl(Γ(1/4)bigr).
We define C_7a to be the irrationality measure of π: C_7a := sup_μ∈ℝ μ such that lvert π - p/q rvert < q^-μ for infinitely many rationals p/q. Equivalently, C_7a is the infimum of all ν such that for every ε>0 there exists q_0(ε) with |π-p/q| > 1/q^ν+ε for all integers p and all integers q ≥ q_0(ε).
Let μ be a Borel probability measure on ℝ^n with finite second moments. Its covariance matrix is Cov(μ) :=\ ∫_ℝ^n (x-m)(x-m)^mathsf T dμ(x), m:=∫_ℝ^n x dμ(x). ### Convex bodies If K⊂ℝ^n is a convex body, let λ_K be the uniform probability measure on K and abbreviate Cov(K):=Cov(λ_K).
C_20c is the KLS constant (Kannan–Lovász–Simonovits constant) for log-concave measures. It is defined as C_20c := sup_n≥ 1 ψ_n, where ψ_n is the worst-case inverse Cheeger (isoperimetric) constant among isotropic log-concave probability measures on ℝ^n.
C_11a is the smallest constant such that, for every n≥ 1 and every function f:-1,1^n → ℝ Ebigl|f(x)-Ef(x)bigr| ≤ C_11aE|∇ f|(x), where x=(x_1,…,x_n) is uniform on -1,1^n and |∇ f|(x)=Bigl(Σ_j=1^n |D_j f(x)|^2Bigr)^1/2, D_j f(x)=f(x)-f(x^(j))/2, with x^(j)=(x_1,...,x_j-1,-x_j,x_j+1,...,x_n).
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.
C_10 is the real Grothendieck constant K_G^ℝ. It is the smallest constant C such that for every m,n ≥ 1 and every real matrix A=(a_ij) ∈ ℝ^m× n one has max_substacku_1,…,u_m, v_1,…,v_n ∈ S^∞ Σ_i=1^m Σ_j=1^n a_ij ⟨ u_i, v_j⟩ ≤ C max_ε_1,…,ε_m, δ_1,…,δ_n = ± 1 Σ_i=1^m Σ_j=1^n a_ij ε_i δ_j.
Let X be a random vector in ℝ^n with an isotropic log-concave distribution (i.e. X has a log-concave density, E X=0, and Cov(X)=Id). Since X is isotropic, E|X|^2 = n.
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?
