On globally sparse Ramsey graphs

We say that a graph \(G\) has the Ramsey property w.r.t.\ some graph \(F\) and some integer \(r\geq 2\), 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{ń}ski asked how globally sparse \((F,r)\)-Ramsey graphs \(G\) can possibly be, where the density of \(G\) is measured by the subgraph \(H\subseteq G\) 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 \(r\geq 2\) colors are available.