In this dissertation we treat three tiling problems and three problems in combinatorial geometry, extremal graph theory and sparse Ramsey theory.

We first consider tilings of $Zn$. In this setting a tile $T$ is just a finite subset of $Zn$. We say that $T$ tiles $Zn$ if the latter set admits a partition into isometric copies of $T$. Chalcraft observed that there exist $T$ that do not tile $Zn$ but tile $Zd$ for some $d>n$. He conjectured that such $d$ exists for any given tile. We prove this conjecture in Chapter 2.

In Chapter 3 we prove a conjecture of Lonc, stating that for any poset $P$ of size a power of $2$, if $P$ has a greatest and a least element, then there is a positive integer $k$ such that $[2]k$ can be partitioned into copies of $P$.

The third tiling problem is about vertex-partitions of the hypercube graph $Qn$. Offner asked: if $G$ is a subgraph of $Qn$ such $|G|$ is a power of $2$, must $V(Qd)$, for some $d$, admit a partition into isomorphic copies of $G$? In Chapter 4 we answer this question in the affirmative.

We follow up with a question in combinatorial geometry. A line in a planar set $P$ is a maximal collinear subset of $P$. P'or and Wood considered colourings of finite $P$ without large lines with a bounded number of colours. In particular, they examined whether monochromatic lines always appear in such colourings provided that $|P|$ is large. They conjectured that for all $k,l\ge 2$ there exists an $n\ge 2$ such that if $|P|\ge n$ and $P$ does not contain a line of cardinality larger than $l$, then every colouring of $P$ with $k$ colours produces a monochromatic line. In Chapter 5 we construct arbitrarily large counterexamples for the case $k=l=3$.

We follow up with a problem in extremal graph theory. For any graph, we say that a given edge is triangular if it forms a triangle with two other edges. How few triangular edges can there be in a graph with $n$ vertices and $m$ edges? For sufficiently large $n$ we prove a conjecture of F"uredi and Maleki that gives an exact formula for this minimum. This proof is given in Chapter 6.

Finally, Chapter 7 is concerned with degrees of vertices in directed hypergraphs. One way to prescribe an orientation to an $r$-uniform graph $H$ is to assign for each of its edges one of the $r!$ possible orderings of its elements. Then, for any $p$-set of vertices $A$ and any $p$-set of indices $I\subset [r]$, we define the $I$-degree of $A$ to be the number of edges containing vertices $A$ in precisely the positions labelled by $I$. Caro and Hansberg were interested in determining whether a given $r$-uniform hypergraph admits an orientation where every set of $p$ vertices has some $I$-degree equal to $0$. They conjectured that a certain Hall-type condition is sufficient. We show that this is true for $r$ large, but false in general.