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
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]
Publication Title
Mathematics
Volume
10
Issue
2
Publisher
MDPI
Publisher Location
ST ALBAN-ANLAGE 66, CH-4052 BASEL, SWITZERLAND