Minimum number of colours to avoid k-term monochromatic arithmetic progressions

Sim, Kai An and Wong, Kok Bin (2022) Minimum number of colours to avoid k-term monochromatic arithmetic progressions. Mathematics, 10 (2). ISSN 2227-7390, DOI

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By recalling van der Waerden theorem, there exists a least a positive integer w=w(k;r) such that for any n & GE;w, every r-colouring of 1,n] admits a monochromatic k-term arithmetic progression. Let k & GE;2 and r(k)(n) denote the minimum number of colour required so that there exists a r(k)(n)-colouring of 1,n] that avoids any monochromatic k-term arithmetic progression. In this paper, we give necessary and sufficient conditions for r(k)(n+1)=r(k)(n). We also show that r(k)(n)=2 for all k & LE;n & LE;2(k-1)(2) and give an upper bound for r(p)(p(m)) for any prime p & GE;3 and integer m & GE;2.

Item Type: Article
Funders: Fundamental Research Grant Scheme (FRGS), Malaysia Ministry of Higher Education and Publication Support Scheme by Sunway University, Malaysia [FRGS/1/2020/STG06/SYUC/03/1]
Uncontrolled Keywords: van der Waerden theorem; Monochromatic arithmetic progression
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science > Institute of Mathematical Sciences
Depositing User: Ms. Juhaida Abd Rahim
Date Deposited: 29 Jul 2022 00:27
Last Modified: 29 Jul 2022 00:27

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