Sim, Kai An and Tan, Ta Sheng and Wong, Kok Bin
(2021)
*A note on the mixed van der Waerden number.*
Bulletin of the Korean Mathematical Society, 58 (6).
pp. 1341-1354.
ISSN 1015-8634,
DOI https://doi.org/10.4134/BKMS.b200718.

## Abstract

Let r >= 2, and let k(i) >= 2 for 1 <= i <= r. Mixed van der Waerden's theorem states that there exists a least positive integer w = w(k(1), k(2), k(3), ..., k(r); r) such that for any n >= w, every r-colouring of 1, n] admits a k(i)-term arithmetic progression with colour i for some i is an element of 1, r]. For k >= 3 and r >= 2, the mixed van der Waerden number w(k, 2, 2, ..., 2; r) is denoted by w(2)(k; r). B. Landman and A. Robertson 9] showed that for k < r < 3/2 (k - 1) and r >= 2k + 2, the inequality w(2)(k; r) <= r(k - 1) holds. In this note, we establish some results on w(2)(k; r) for 2 <= r <= k.

Item Type: | Article |
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Funders: | Fundamental Research Grant Scheme (FRGS) (FRGS/1/2020/STG06/SYUC/03/1), Ministry of Education, Malaysia |

Uncontrolled Keywords: | Mixed van der Waerden number; Ramsey theory on the integers |

Subjects: | Q Science > QA Mathematics |

Divisions: | Faculty of Science > Institute of Mathematical Sciences |

Depositing User: | Ms Zaharah Ramly |

Date Deposited: | 25 May 2022 07:33 |

Last Modified: | 25 May 2022 07:33 |

URI: | http://eprints.um.edu.my/id/eprint/27156 |

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