Rainbow cliques in edge-colored graphs

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