  An ordered graph $G$ is a graph together with a specified linear ordering on
the vertices, and its interval chromatic number is the minimum number of
independent sets consisting of consecutive vertices that are needed to
partition the vertex set. The $t$-color Ramsey number $R_t(G)$ of an ordered
graph $G$ is the minimum number of vertices of an ordered complete graph such
that every edge-coloring from a set of $t$ colors contains a monochromatic copy
of $G$ such that the copy of $G$ preserves the original ordering on $G$. An
ordered graph is $k$-ichromatic if it has interval chromatic number $k$. We
obtain lower bounds linear in the number of vertices for the Ramsey numbers of
certain classes of 2-ichromatic ordered graphs. We also determine the exact
value of the $t$-color Ramsey number for two families of 2-ichromatic ordered
graphs, and we prove a linear upper bound for a class of 2-ichromatic ordered
graphs.
