Multipartite graph-sparse graph Ramsey numbers (Q1076043)

Let \(F\) and \(G\) be finite graphs. The Ramsey number \(r(F,G)\) is the smallest positive integer n so that, given any graph on \(n\) vertices, either it contains a subgraph isomorphic to \(F\) or its complement contains a subgraph isomorphic to \(G\). In this paper, the Ramsey number \(r(F,G)\) is determined in the case where \(F\) is an arbitrary fixed graph and \(G\) is a sufficiently large sparse connected graph with restrictions on the maximum degree of its vertices. An asymptotically correct upper bound is obtained for \(f(F,T)\) where \(T\) is a sufficiently large, but otherwise arbitrary, tree.