Ramsey for complete graphs with a dropped edge or a triangle

We first present a general upper bound (containing the well-known Graham-Rödl upper bound for complete graphs in the particular case when ti=1 for all i). We then focus our attention to when r=2, and to dropped cliques of order 2 and 3 (edges and triangles). We give the exact value for R(K[n,2],K[4,3]) and R(K[n,3],K[4,3]) for all n≥2.