# Into Darkness

Two things are infinite: the universe and human stupidity; and I’m not sure about the universe.
― Albert Einstein

#### Combinatorics

• (The linear arboricity conjecture) The linear arboricity of every $d$-regular graph is $\lceil (d+1)/2 \rceil$.
• (Packing squares) Let $s(x)$ denote the maximum number of non-overlapping unit squares which can be packed into a large square of side length $x$. Find the best asymptotics of the “wasted” area $W(x)=x^2-s(x)$.
• (Graham’s conjecture on sub-multiplicativity of pebbling number) $\pi(G_1\times G_2) \le \pi(G_1)\pi(G_2)$.

#### Discrete Geometry

• (Erdös-Szekeres problem) For any integer $n\geq 3$, any set of at least $2^{n-2}+1$ points in general position in the plane contains $n$ points that are the vertices of a convex $n$-gon.
• (Makai–Pach’s conjecture on translative slab covering) A sequence of slabs (region enclosed by two parallel hyperplanes) in $\mathbb{R}^d$ with widths (distance between two hyperplanes) $w_1, w_2, \dots$ permits a translative covering if $\sum_i w_i = \infty$.

#### Computability

• (Freeness of matrix semigroup) Is it decidable whether the multiplicative semigroup generated by two $2\times 2$ matrices in the modular group $SL_2(\mathbb{Z})$ is free?

#### Metric Geometry

• (Inequality of the Means) Is is possible to pack $n^n$ rectangular $n$-dimensional boxes each of which has side lengths $a_1,a_2,\ldots,a_n$ inside an $n$-dimensional cube with side length $a_1 + a_2 + \ldots a_n$?
• (Michael Atiyah) Let $x_1, \ldots, x_n$ be $n$ distinct points in the unit ball. Let the oriented line $x_ix_j$ meet the boundary 2-sphere in a point $t_{ij}$ (regarded as a point of the complex Riemann sphere $\mathbb{C}\cup\infty$). For the complex polynomial $p_i$ of degree $n-1$, whose roots are $t_{ij}$. For all $(x_1, \ldots, x_n)$ the $n$ polynomials are linearly independent.

#### Number Theory

• (Lehmer’s conjecture) Is there an absolute constant $\mu > 1$ such that for every monic polynomial $p(x)$ with integer coefficients, the product of the roots of $p(x)$ with length greater than $1$ is greater than $\mu$?

#### Topology

• Is there a continuous mapping $f: [0,1]\to\mathbb{R}^2$ such that for all $\epsilon>0$, the image of $[0,\epsilon]$ under $f$ is a convex set with three points not on a line?