New lower bounds for van der Waerden numbers
We show that there is a red-blue colouring of $[N]$ with no blue 3-term arithmetic progression and no red arithmetic progression of length $e^{C(\log N)^{3/4}(\log \log N)^{1/4}}$. Consequently, the two-colour van der Waerden number $w(3,k)$ is bounded below by $k^{b(k)}$, where $b(k) = c \big ( \fr...
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| Format: | Article |
| Language: | English |
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Cambridge University Press
2022-01-01
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| Series: | Forum of Mathematics, Pi |
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| Online Access: | https://www.cambridge.org/core/product/identifier/S2050508622000129/type/journal_article |
| _version_ | 1811156241989238784 |
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| author | Ben Green |
| author_facet | Ben Green |
| author_sort | Ben Green |
| collection | DOAJ |
| description | We show that there is a red-blue colouring of $[N]$ with no blue 3-term arithmetic progression and no red arithmetic progression of length $e^{C(\log N)^{3/4}(\log \log N)^{1/4}}$. Consequently, the two-colour van der Waerden number $w(3,k)$ is bounded below by $k^{b(k)}$, where $b(k) = c \big ( \frac {\log k}{\log \log k} \big )^{1/3}$. Previously it had been speculated, supported by data, that $w(3,k) = O(k^2)$. |
| first_indexed | 2024-04-10T04:48:20Z |
| format | Article |
| id | doaj.art-81602d7165b84b0ebd3cab6abf8aacba |
| institution | Directory Open Access Journal |
| issn | 2050-5086 |
| language | English |
| last_indexed | 2024-04-10T04:48:20Z |
| publishDate | 2022-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Pi |
| spelling | doaj.art-81602d7165b84b0ebd3cab6abf8aacba2023-03-09T12:34:22ZengCambridge University PressForum of Mathematics, Pi2050-50862022-01-011010.1017/fmp.2022.12New lower bounds for van der Waerden numbersBen Green0https://orcid.org/0000-0002-2224-1193Mathematical Institute, University of Oxford, Andrew Wiles Building, Woodstock Rd, Oxford, OX2 6RE, UK;We show that there is a red-blue colouring of $[N]$ with no blue 3-term arithmetic progression and no red arithmetic progression of length $e^{C(\log N)^{3/4}(\log \log N)^{1/4}}$. Consequently, the two-colour van der Waerden number $w(3,k)$ is bounded below by $k^{b(k)}$, where $b(k) = c \big ( \frac {\log k}{\log \log k} \big )^{1/3}$. Previously it had been speculated, supported by data, that $w(3,k) = O(k^2)$.https://www.cambridge.org/core/product/identifier/S2050508622000129/type/journal_article11B2511B30 |
| spellingShingle | Ben Green New lower bounds for van der Waerden numbers Forum of Mathematics, Pi 11B25 11B30 |
| title | New lower bounds for van der Waerden numbers |
| title_full | New lower bounds for van der Waerden numbers |
| title_fullStr | New lower bounds for van der Waerden numbers |
| title_full_unstemmed | New lower bounds for van der Waerden numbers |
| title_short | New lower bounds for van der Waerden numbers |
| title_sort | new lower bounds for van der waerden numbers |
| topic | 11B25 11B30 |
| url | https://www.cambridge.org/core/product/identifier/S2050508622000129/type/journal_article |
| work_keys_str_mv | AT bengreen newlowerboundsforvanderwaerdennumbers |