# 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?

Euclidean 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?
42 (this post is made with love)