Constants

Brackets in a cell's corner mark bounds verified by an automatically checked certificate: [ lower, ] upper. Green matches the best bound in the literature, purple is stronger than (or absent from) it, and gray is weaker. See the bounds that no certificate matches yet.

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03D Computability and recursion theory

Busy beaver ones function

The busy beaver ones function $\Sigma(n)$ (Radó’s sigma function) is the maximum number of 1s left on the tape by an $n$-state, 2-symbol Turing machine that halts when started on a blank tape.

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Busy beaver shift function

The busy beaver shift function $BB(n)$, also written $S(n)$, is the maximum number of steps made by an $n$-state, 2-symbol Turing machine that halts when started on a blank tape.

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05A Enumerative combinatorics

Map folding

The number of distinct ways to fold an $m \times n$ rectangle into a unit square.

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Minimal superpermutation problem

$L(n)$ is the length of the shortest string that contains each permutation of $n$ symbols as a contiguous substring.

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Stamp folding

The number of distinct ways to fold a strip of $n$ labeled stamps.

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05B Designs and configurations

Optimal Golomb rulers

A Golomb ruler is a set of integer marks such that no two pairs of marks are the same distance apart. $G(n)$ is the shortest length of a Golomb ruler with $n$ marks.

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05C Graph theory

Cages

$n(k,g)$ is the order of a $(k,g)$-cage, a $k$-regular graph of girth $g$ with the fewest possible vertices.

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Coil-in-the-box

A coil in the $n$-dimensional hypercube is an induced cycle: a cycle whose vertices are adjacent only when they’re consecutive around the cycle. $c(n)$ is the length, in edges, of the longest coil in the $n$-dimensional hypercube.

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Degree-diameter problem

$N(d,k)$ is the largest possible number of vertices in a graph of maximum degree $d$ and diameter at most $k$.

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Snake-in-the-box

A snake in the $n$-dimensional hypercube is an induced path: a path whose vertices are adjacent only when they’re consecutive on the path. $s(n)$ is the length, in edges, of the longest snake in the $n$-dimensional hypercube.

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05D Extremal combinatorics

Ramsey numbers

The Ramsey number $R(r,s)$ is the smallest number $n$ such that any two-coloring of the edges of the complete graph $K_n$ must contain either a monochromatic $K_r$ in the first color or a monochromatic $K_s$ in the second color.

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Schur numbers

The Schur number $S(k)$ is the largest number $n$ such that the integers $1$ to $n$ can be colored with $k$ colors with no monochromatic solution to $x + y = z$.

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Van der Waerden numbers

$W(r,k)$ is the smallest number $n$ such that if the integers $1$ to $n$ are colored with $r$ colors, there must be a monochromatic arithmetic progression of length $k$.

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Weak Schur numbers

The weak Schur number $WS(k)$ is the largest number $n$ such that the integers $1$ to $n$ can be colored with $k$ colors with no monochromatic solution to $x + y = z$ with $x \neq y$.

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11M Zeta and L-functions: analytic theory

de Bruijn–Newman constant

The de Bruijn–Newman constant $\Lambda$ is defined via the zeros of a certain function $H(\lambda,z)$, where $\lambda$ is a real parameter and $z$ is a complex variable. More precisely, $H(\lambda, z):=\int_{0}^{\infty} e^{\lambda u^{2}} \Phi(u) \cos (z u) \, du$, where $\Phi$ is the super-exponentially decaying function $\Phi(u) = \sum_{n=1}^{\infty} (2\pi^2n^4e^{9u}-3\pi n^2 e^{5u} ) e^{-\pi n^2 e^{4u}}$ and $\Lambda$ is the unique real number with the property that $H(\lambda, \cdot)$ has only real zeros if and only if $\lambda\geq \Lambda$. The Riemann hypothesis is equivalent to $\Lambda \le 0$.

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52C Discrete geometry

Arrangements of lines in the affine plane

The number of ways of arranging $n$ lines in the (affine) plane.

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Arrangements of lines in general position

The number of ways of arranging $n$ straight lines in general position in the (affine) plane.

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Circle packing in an equilateral triangle

The side length of the smallest equilateral triangle into which $n$ unit circles (of radius 1) can be packed.

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Disk covering problem

The smallest value $r(n)$ such that $n$ disks of radius $r(n)$ can be arranged to cover the unit disk.

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Kissing numbers

The kissing number of $\mathbb{R}^n$ is the maximum number of non-overlapping unit spheres that can touch a central unit sphere in $\mathbb{R}^n$.

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Packing unit squares in a square

$s(n)$ is the side of the smallest square into which $n$ unit squares can be packed.

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