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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 graphdoi:10.1016/j.disc.2011.09.015 fatcat:gjjizn7b6zaqzjeypu5plj5tom