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An edge-coloured graph is said to be rainbow if no colour appears more thanonce. Extremal problems involving rainbow objects have been a focus of muchresearch over the last decade as they capture the essence of a number ofinteresting problems in a variety of areas. A particularly intensively studiedquestion due to Keevash, Mubayi, Sudakov and Verstra\"ete from 2007 asks forthe maximum possible average degree of a properly edge-coloured graph on $n$vertices without a rainbow cycle. Improving upon a series of earlier bounds,Tomon proved an upper bound of $(\log n)^{2+o(1)}$ for this question. Veryrecently, Janzer-Sudakov and Kim-Lee-Liu-Tran independently removed the $o(1)$term in Tomon's bound, showing a bound of $O(\log^2 n)$. We prove an upperbound of $(\log n)^{1+o(1)}$ for this maximum possible average degree whenthere is no rainbow cycle. Our result is tight up to the $o(1)$ term, and so itessentially resolves this question. In addition, we observe a connectionbetween this problem and several questions in additive number theory, allowingus to extend existing results on these questions for abelian groups to the caseof non-abelian groups.

Essentially tight bounds for rainbow cycles in proper edge-colourings