5 problems

Let $T$ be a tile in $\mathbb{Z}^n$. A tiling by translated copies of $T$ is periodic if it is invariant under a nonzero translation of $\mathbb{Z}^n$. Periodic tiling conjecture.…

A rectangle tiling set is a finite set $R$ of rectangles, and a simply connected region is a region in the plane without holes that is to be tiled using translated copies of rectan…

Let $R$ be an $n\times n$ expanding integer matrix and let $0\in\mathcal D\subset\mathbb Z^n$ satisfy $\#\mathcal D=|\det(R)|$. Define $T(R,\mathcal D)$ to be the associated self-a…

Let $p$ be a prime and $d\geq 1$. A set $E\subseteq (\mathbb Z/p\mathbb Z)^d$ is tiling if there exists $T\subseteq (\mathbb Z/p\mathbb Z)^d$ such that the translates…

Let $M$ be a standard multigraph on $n$ vertices, where $n$ is divisible by $3$. Let $\delta(M)$ denote its minimum degree, let $H_M$ be its heavy-edge graph, and let a $T_5$-facto…