We say that a graph G has the Ramsey property w.r.t.\ some graph F and
some integer r≥2, or G is (F,r)-Ramsey for short, if any r-coloring
of the edges of G contains a monochromatic copy of F. R{\"o}dl and
Ruci{\'n}ski asked how globally sparse (F,r)-Ramsey graphs G can possibly
be, where the density of G is measured by the subgraph H⊆G with
the highest average degree. So far, this so-called Ramsey density is known only
for cliques and some trivial graphs F. In this work we determine the Ramsey
density up to some small error terms for several cases when F is a complete
bipartite graph, a cycle or a path, and r≥2 colors are available