Essentially tight bounds for rainbow cycles in proper edge-colourings

An edge-coloured graph is said to be rainbow if no colour appears more than once. Extremal problems involving rainbow objects have been a focus of much research over the last decade as they capture the essence of a number of interesting problems in a variety of areas. A particularly intensively studied question due to Keevash, Mubayi, Sudakov and Verstraëte from 2007 asks for the maximum possible average degree of a properly edge-coloured graph on (Formula presented.) vertices without a rainbow cycle. Improving upon a series of earlier bounds, Tomon proved an upper bound of (Formula presented.) for this question. Very recently, Janzer–Sudakov and Kim–Lee–Liu–Tran independently removed the (Formula presented.) term in Tomon's bound, showing a bound of (Formula presented.). We prove an upper bound of (Formula presented.) for this maximum possible average degree when there is no rainbow cycle. Our result is tight up to the (Formula presented.) term, and so, it essentially resolves this question. In addition, we observe a connection between this problem and several questions in additive number theory, allowing us to extend existing results on these questions for abelian groups to the case of non-abelian groups.