A note on the mixed van der Waerden number
Document Type
Article
Publication Date
1-1-2021
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.
Keywords
Mixed van der Waerden number, Ramsey theory on the integers
Divisions
MathematicalSciences
Funders
Fundamental Research Grant Scheme (FRGS) (FRGS/1/2020/STG06/SYUC/03/1),Ministry of Education, Malaysia
Publication Title
Bulletin of the Korean Mathematical Society
Volume
58
Issue
6
Publisher
Korean Mathematical Society
Publisher Location
KOREA SCIENCE TECHNOLOGY CTR 202, 635-4 YEOKSAM-DONG, KANGNAM-KU, SEOUL 135-703, SOUTH KOREA