Destroying the Ramsey property by the removal of edges

For any two graphs G and H, the Ramsey number R(G, H) is the minimum number of vertices required in a complete graph to guarantee that every red/blue coloring of the edges of that complete graph contains either a red subgraph isomorphic to G or a blue subgraph isomorphic to H. Hence, the removal of a single vertex in the complete graph destroys this property. Rather than remove a vertex (along with all of its incident edges), we consider the problem of selecting a vertex and removing edges incident with it. Our goal is to determine, for various pairs of graphs G and H, the exact number of edges that must be removed in this way in order to destroy the Ramsey property. We give precise evaluations of this number in the cases where G is a tree and H is a complete graph and in the cases where G and H are both stars. Partial results are obtained in other cases in which G and H are trees, not both of which are stars.