  An ordered graph $\mathcal{G}$ is a simple graph together with a total
ordering on its vertices. The (2-color) Ramsey number of $\mathcal{G}$ is the
smallest integer $N$ such that every 2-coloring of the edges of the complete
ordered graph on $N$ vertices has a monochromatic copy of $\mathcal{G}$ that
respects the ordering. In this paper we investigate the effect of various graph
operations on the Ramsey number of a given ordered graph, and detail a general
framework for applying results on extremal functions of 0-1 matrices to ordered
Ramsey problems. We apply this method to give upper bounds on the Ramsey number
of ordered matchings arising from sum-decomposable permutations, an alternating
ordering of the cycle, and an alternating ordering of the tight hyperpath. We
also construct ordered matchings on $n$ vertices whose Ramsey number is
$n^{q+o(1)}$ for any given exponent $q\in(1,2)$.
