Some Ramsey–Turán type problems and related questions

In this paper the following Ramsey–Turán type problem is one of several addressed. For which graphs G does there exist a constant 0 < c < 1 such that when H is a graph of order the Ramsey number r ( G ) with δ ( H ) > c | H | , then any 2-edge coloring of H contains a monochromatic copy of G ? Specific results, conjectures, and questions with suggested values for c are considered when G is an odd cycle, path, or tree of limited maximum degree. Another variant is to 2-edge color a replacement for the graph K r ( G ) by a balanced multipartite graph of approximately the same order with the same consequence, a monochromatic G .