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 https://doi.org/10.3390/math10020247.

## Abstract

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 |
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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 |

URI: | http://eprints.um.edu.my/id/eprint/33599 |

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