Essentially tight bounds for rainbow cycles in proper edge-colourings (Matija Bucic, Princeton)

Combinatorics Seminar
Friday, March 1, 2024 - 3:15pm for 1 hour (actually 50 minutes)
Skiles 308
Matija Bucic – Princeton University – mb5225@princeton.edu
Evelyne Smith-Roberge

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