## Theory of Combinatorial Algorithms

Prof. Emo Welzl and Prof. Bernd Gärtner

# Mittagsseminar (in cooperation with M. Ghaffari, A. Steger and B. Sudakov)

 Mittagsseminar Talk Information

Date and Time: Tuesday, May 30, 2017, 12:15 pm

Duration: 30 minutes

Location: CAB G51

Speaker: Vytautas Gruslys (University of Cambridge)

## Tilings of high-dimensional spaces

Any Tetris tile is a connected subset of $\mathbb{Z}^2$ of size $4$. More generally, we define an $n$-dimensional shape to be any (non-empty) finite subset of $\mathbb{Z}^n$. Does a given shape $T \subset \mathbb{Z}^n$ tile the $n$-dimensional space, meaning that $\mathbb{Z}^n$ can be partitioned into copies of $T$? Of course, some shapes tile $\mathbb{Z}^n$ and some do not. Moreover, some shapes that do not tile $\mathbb{Z}^n$ do tile $\mathbb{Z}^{n+1}$. Chalcraft conjectured that every shape in $\mathbb{Z}^n$ tiles $\mathbb{Z}^d$ for some $d \ge n$. We prove this conjecture and examine related questions regarding posets and the hypercube graph $Q_n$. This talk is based on joint work with Leader, Tan and Tomon.

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