  Two independent edges in ordered graphs can be nested, crossing or separated.
These relations define six types of subgraphs, depending on which relations are
forbidden. We refine a remark by Erd\H{o}s and Rado that every 2-coloring of
the edges of a complete graph contains a monochromatic spanning tree. We show
that forbidding one relation we always have a monochromatic (non-nested,
non-crossing, non-separated) spanning tree in a 2-edge-colored ordered complete
graph. On the other hand, if two relations are forbidden, then it is possible
that we have monochromatic (nested, separated, crossing) subtrees of size only
half the number of vertices. The existence of a monochromatic non-nested
spanning tree in 2-colorings of ordered complete graphs verifies a more general
conjecture for twisted drawings. Our second subject is to refine the Ramsey
number of matchings for ordered complete graphs. Cockayne and Lorimer proved
that for given positive integers t, n, m=(t-1)(n-1)+2n is the smallest integer
with the following property: every t-coloring of the edges of a complete graph
Km contains a monochromatic matching with n edges. We conjecture a
strengthening: t-colored ordered complete graphs on m vertices contain
monochromatic non-nested and also non-separated matchings with n edges. We
prove this conjecture for some special cases. (i) Every t-colored ordered
complete graph on t+3 vertices contains a monochromatic non-nested matching of
size two. (ii) Every 2-colored ordered complete graph on 3n-1 vertices contains
a monochromatic non-separated matching of size n.
  For nested, separated, and crossing matchings the situation is different. The
smallest m ensuring a monochromatic matching of size n in every t-coloring is
2(t(n-1))+1) in the first two cases and one less in the third case.
