Exact values of defective Ramsey numbers in graph classes

Given a graph $G$, a $k$-sparse $j$-set is a set of $j$ vertices inducing a subgraph with maximum degree at most $k$. A $k$-dense $i$-set is a set of $i$ vertices that is $k$-sparse in the complement of $G$. As a generalization of Ramsey numbers, the $k$-defective Ramsey number $R_k^G(i,j)$ for the graph class $G$ is defined as the smallest natural number $n$ such that all graphs on $n$ vertices in the class $G$ have either a $k$-dense $i$-set or a $k$-sparse $j$-set. In this paper, we examine $R_k^G(i,j)$ where $G$ represents various graph classes. For forests and cographs, we give exact formulas for all defective Ramsey numbers. For cacti, bipartite graphs and split graphs, we derive defective Ramsey numbers in most of the cases and point out open questions, formulated as conjectures if possible.