Rainbow Turán problem for even cycles

An edge-colored graph is rainbow if all its edges are colored with distinct colors. For a fixed graph H , the rainbow Turán number ex(n,H) is defined as themaximumnumber of edges in a properly edge-colored graph on n vertices with no rainbow copy of H . We study the rainbow Turán number of even cycles, and prove that for every fixed ε > 0, there is a constant C(ε) such that every properly edge-colored graph on n vertices with at least C(ε)n1+ε edges contains a rainbow cycle of even length at most 2  ln 4−ln ε ln(1+ε)  . This partially answers a question of Keevash,Mubayi, Sudakov, and Verstraëte, who asked how dense a graph can be without having a rainbow cycle of any length. © 2013 Elsevier Ltd. All rights reserved.