  Gallai-colorings are edge-colored complete graphs in which there are no
rainbow triangles. Within such colored complete graphs, we consider Ramsey-type
questions, looking for specified monochromatic graphs. In this work, we
consider monochromatic bipartite graphs since the numbers are known to grow
more slowly than for non-bipartite graphs. The main result shows that it
suffices to consider only $3$-colorings which have a special partition of the
vertices. Using this tool, we find several sharp numbers and conjecture the
sharp value for all bipartite graphs. In particular, we determine the
Gallai-Ramsey numbers for all bipartite graphs with two vertices in one part
and initiate the study of linear forests.
