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 ≥ 2 , or G is ( F , r ) -Ramsey for short, if any r -coloring of the edges of G contains a monochromatic copy of F . Rödl and Ruciński 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.