Rainbow Turلn 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 the maximum number 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 ) n 1 + 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.