A note on the mixed van der Waerden number

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.

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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
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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