  An ordered graph $H$ on $n$ vertices is a graph whose vertices have been
labeled bijectively with $\{1,...,n\}$. The ordered Ramsey number $r_<(H)$ is
the minimum $n$ such that every two-coloring of the edges of the complete graph
$K_n$ contains a monochromatic copy of $H$ such that the vertices in the copy
appear in the same order as in $H$.
  Although some bounds on the ordered Ramsey numbers of certain infinite
families of graphs are known, very little is known about the ordered Ramsey
numbers of specific small graphs compared to how much we know about the usual
Ramsey numbers for these graphs. In this paper we tackle the problem of proving
non-trivial upper bounds on orderings of graphs on four vertices. We also
extend one of our results to $n+1$ vertex graphs that consist of a complete
graph on $n$ vertices with a pendant edge to vertex 1. Finally, we use a SAT
solver to compute some numbers exactly.
