  An edge coloring of the n-vertex complete graph K_n is a Gallai coloring if
it does not contain any rainbow triangle, that is, a triangle whose edges are
colored with three distinct colors. We prove that the number of Gallai
colorings of K_n with at most three colors is at most 7(n+1)*2^{n choose 2},
which improves the best known upper bound of \frac{3}{2} * (n-1)! * 2^{(n-1)
choose 2} in [Discrete Mathematics, 2017].
