AbstractChvátal established thatr(Tm, Kn) = (m– 1)(n– 1) + 1, whereTmis an arbitrary tree of ordermandKnis the complete graph of ordern. This result was extended by Chartrand, Gould, and Polimeni who showedKncould be replaced by a graph with clique numbernand ordern+ 1 providedn≧ 3 andm≧ 3. We further extend these results to show thatKncan be replaced by any graph onn+ 2 vertices with clique numbern, providedn≧ 5 andm≧ 4. We then show that further extensions, in particular to graphs onn+ 3 vertices with clique numbernare impossible. We also investigate the Ramsey number of trees versus complete graphs minus sets of independent edges. We show thatr(Tm, Kn–tK2) = (m– 1)(n – t– 1) + 1 form≧ 3,n≧ 6, whereTmis any tree of ordermexcept the star, and for