On October 17, 2023, Matija Bucić from Princeton University gave a talk at the Discrete Math Seminar on the edge density of graphs with a proper coloring without a rainbow cycle. The title of his talk was “Essentially tight bounds for rainbow cycles in proper edge-colourings“.

## Matija Bucić, Essentially tight bounds for rainbow cycles in proper edge-colourings

An edge-coloured graph is said to be rainbow if it uses no colour 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 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. Very recently, Janzer-Sudakov and Kim-Lee-Liu-Tran independently removed the $o(1)$ term in Tomon’s bound. We show that the answer to the question is equal to $(\log n)^{1+o(1)}$.

A key tool we use is the theory of robust sublinear expanders. 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.

Joint work with: Noga Alon, Lisa Sauermann, Dmitrii Zakharov and Or Zamir.