On Generalized Ramsey Numbers

In this thesis,we state the history of research on classical Ramsey numbers and generalized Ramsey numbers.Given two simple graphs F and G,the generalized Ramsey number R(F,G)is the smallest integer N for which every red-blue edge coloring of the complete graph KN produces either a red F or a blue G.A connected graph G of order n is called 3-good if R(K3,G)=2n-1.Let f(n)denote the largest integer q such that every graph G of order n and size q is 3-good.In 1980,Burr,Erdos,Faudree,Rousseau,and Schelp proved that f(n)?(?)?1.13n+0.07 for all n?4 and f(n)<(27/4+o(1))n(log n)2 for sufficiently large n.There is no improvement in decades.Using the approach of Burr et al.,we obtain a better lemma and improve the bounds of f(n).We prove that f(n)? n+(?)?1.16n-0.08 for all n?4 and f(n)<(1+o(1))n log n for sufficiently large n.