## Combinatorics Colloquium

- Event Type
- Seminar/Symposium
- Sponsor
- Department of Mathematics
- Location
- Burrill Hall 124
- Date
- Aug 29, 2024 1:00 pm
- Speaker
- Matija Bucić (Princeton)
- Contact
- Peter Bradshaw
- pb38@illinois.edu
- Views
- 20
- Originating Calendar
- Combinatorics Research Area Calendar
Speaker: Matija Bucić

Title: Essentially tight bounds for rainbow cycles in proper edge-colourings

Abstract: 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.