Deleting edges from Ramsey graphs

Abstract

For graphs F, G and H, let F→(G,H) signify that any red–blue edge coloring of F contains either a red G or a blue H. Define the Ramsey number R(G,H) to be the smallest r such that Kr→(G,H), and the complete-critical Ramsey number RK(G,H) to be the largest p such that Kr∖Kp→(G,H), where r=R(G,H). In this note, we shall show a general upper bound for RK(G,H) and determine the exact values of RK(K1,n,Km), RK(Pn,C4) and RK(Fn,K3), respectively.