Some Ramsey-Turan type problems and related questions

In this paper the following Ramsey-Turan 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 delta(H) > c vertical bar H vertical bar, 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. (C) 2011 Elsevier B.V. All rights reserved.