The $k$-color Ramsey number of the complete graph $K_n$, denoted with $R_k(n)$, is defined to be the smallest integer $t$, such that in any $k$-coloring of the edges of $K_t$, there is a complete subgraph $K_n$ all of whose edges have the same color.

I'm looking for results (**if exist**) that link together Ramsey numbers of increasing number of colors; in particular, **constructive** ways to prove lower bounds.

For example: suppose that we have an instance of $K_{r_1}$ that proves $R_{k_1}(n_1) \gt r_1$ then we can build an instance of $ K_{r_2}$ that proves $R_{k_2}(n_2) \gt r_2$ for some particular $k_2 \lt k_1$ and $n_2 \gt n_1$

Can you give me some results / references?

(or results of the same type that applies to particular class of graphs, for example complete bipartite graphs)