Essentially tight bounds for rainbow cycles in proper edge-colourings
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866913707833950208 |
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| author | Alon, Noga Bucić, Matija Sauermann, Lisa Zakharov, Dmitrii Zamir, Or |
| author_facet | Alon, Noga Bucić, Matija Sauermann, Lisa Zakharov, Dmitrii Zamir, Or |
| contents | 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 $n$ 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, showing a bound of $O(\log^2 n)$. We prove an upper bound of $(\log n)^{1+o(1)}$ for this maximum possible average degree when there is no rainbow cycle. Our result is tight up to the $o(1)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_04460 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Essentially tight bounds for rainbow cycles in proper edge-colourings Alon, Noga Bucić, Matija Sauermann, Lisa Zakharov, Dmitrii Zamir, Or Combinatorics Group Theory Number Theory 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 $n$ 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, showing a bound of $O(\log^2 n)$. We prove an upper bound of $(\log n)^{1+o(1)}$ for this maximum possible average degree when there is no rainbow cycle. Our result is tight up to the $o(1)$ 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. |
| title | Essentially tight bounds for rainbow cycles in proper edge-colourings |
| topic | Combinatorics Group Theory Number Theory |
| url | https://arxiv.org/abs/2309.04460 |