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

Document Type

Article

Publication Date

1-1-2022

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.

Keywords

van der Waerden theorem, Monochromatic arithmetic progression

Publication Title

Mathematics

Divisions

MathematicalSciences

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]

Volume

10

Issue

2

Publisher

MDPI

Publisher Location

ST ALBAN-ANLAGE 66, CH-4052 BASEL, SWITZERLAND

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