Ramsey numbers for set-colorings

For s, t, n ∈ N with s ≥ t, an (s, t)-coloring of K$_n$ is an edge coloring of Kn in which each edge is assigned a set of t colors from {1, . . . , s}. For k ∈ N, a monochromatic K$_k$ is a set of k vertices S such that for some color i ∈ [s], i ∈ c(uv) for all distinct u, v ∈ S. As in the case of the classical Ramsey number, we are interested in the least positive integer n = R$_{s,t}$(k) such that for any (s, t)-coloring of K$_n$, there exists a monochromatic K$_k$. We estimate upper and lower bounds for general cases and calculate close bounds for some small cases of R$_{s,t}$(k).