Let $x\neq \emptyset$ be a set and let $\text{Part}(x)$ be the collection of
all partitions of $x$. We need the following notation. Let $P \in \text{Part}(x)$
and $t\subseteq x$. We set
$$P_{[t]} = \{p\in P:p\cap t \neq \emptyset\}.$$
We define the *tiling relation*
on $\text{Part}(x)$ by
$$ P \triangleleft Q \textrm{ if and only if for all } S\subseteq
P\textrm{ we have } \mathsf{card}(S) \leq \mathsf{card}(Q_{[\bigcup S]}).$$
In other words, the relation $P\triangleleft Q$ holds if no subset $S$ of $P$
is covered by a subset of $Q$ having a smaller cardinality than $S$.

**Questions.**

- Is the relation $\triangleleft$ on $\text{Part}(x)$ transitive?
- For $P,Q \in \text{Part}(x)$ is there $Z\in \text{Part}(x)$ such that
- $Z$ refines $P\cup Q$ and
- $Z \triangleleft P$ and $Z\triangleleft Q$?