Rainbow Even Cycles

.We prove that every family of (not necessarily distinct) even cycles \(D_1, , D1.2(n-1) +1\) on some fixed \(n\)-vertex set has a rainbow even cycle (that is, a set of edges from distinct \(D_i\)'s, forming an even cycle). This resolves an open problem of Aharoni, Briggs, Holzman and Jiang. Moreover, the result is best possible for every positive integer \(n\).