Ramsey theory dates back to the 1930's and computing Ramsey numbers is a notoriously
dicult problem in combinatorics. We study Ramsey numbers of graphs under Gallai col-
orings, where a Gallai coloring is a coloring of the edges of a complete graph such that no
triangle has all its edges colored dierently. Given a graph H and an integer k 1, the
Gallai-Ramsey number of H is the least positive integer N such that every Gallai coloring of
the complete graph KN using k colors contains a monochromatic copy of H. Gallai-Ramsey
numbers of graphs are more well-behaved, though computing them is far from trivial. In
this talk, I will present our recent results on Gallai-Ramsey numbers of cycles.