  Much recent progress in hypergraph Ramsey theory has focused on constructions
that lead to lower bounds for the corresponding Ramsey numbers. In this paper,
we consider applications of these results to Gallai colorings. That is, we
focus on the Ramsey numbers resulting from only considering $t$-colorings of
the hyperedges of complete $r$-uniform hypergraphs in which no rainbow
$K_{r+1}^{(r)}$-subhypergraphs exists. We also provide new constructions which
imply improved lower bounds for many $3$ and $4$-uniform Ramsey numbers and $3$
and $4$-uniform Gallai-Ramsey numbers.
