Rainbow cliques in edge-colored graphs

An edge-colored graph H is called rainbow if e(H)=c(H), where e(H) and c(H) are the number of edges of H and colors used in H, respectively. For two graphs G and H, the rainbow number rb(G, H) is the minimum number of colors k such that for every edge-coloring of G using k colors, G contains a rainbow H. In this paper we prove that for an edge-colored graph G on n vertices with n≥k≥4, if e(G)+c(G)≥(n2)+tn,k-2+2, then G contains a rainbow clique K_k, where t_n,k-2 is the Turán number. This implies the known result rb(K_n, K_k)=t_n,k-2+2, and moreover, rb(G,Kk)≤e(G)+rb(Kn,Kk) for n≥k≥4.