  An ordered matching of size $n$ is a graph on a linearly ordered vertex set
$V$, $|V|=2n$, consisting of $n$ pairwise disjoint edges. There are three
different ordered matchings of size two on $V=\{1,2,3,4\}$: an alignment
$\{1,2\},\{3,4\}$, a nesting $\{1,4\},\{2,3\}$, and a crossing
$\{1,3\},\{2,4\}$. Accordingly, there are three basic homogeneous types of
ordered matchings (with all pairs of edges arranged in the same way) which we
call, respectively, lines, stacks, and waves.
  We prove an Erd\H{o}s-Szekeres type result guaranteeing in every ordered
matching of size $n$ the presence of one of the three basic sub-structures of a
given size. In particular, one of them must be of size at least $n^{1/3}$. We
also investigate the size of each of the three sub-structures in a random
ordered matching. Additionally, the former result is generalized to $3$-uniform
ordered matchings.
  Another type of unavoidable patterns we study are twins, that is, pairs of
order-isomorphic, disjoint sub-matchings. By relating to a similar problem for
permutations, we prove that the maximum size of twins that occur in every
ordered matching of size $n$ is $O\left(n^{2/3}\right)$ and
$\Omega\left(n^{3/5}\right)$. We conjecture that the upper bound is the correct
order of magnitude and confirm it for almost all matchings. In fact, our
results for twins are proved more generally for $r$-multiple twins, $r\ge2$.
