Minimum Number of Colours to Avoid k -Term Monochromatic Arithmetic Progressions

Kai An Sim and Kok Bin Wong
Additional contact information
Kai An Sim: School of Mathematical Sciences, Sunway University, Bandar Sunway 47500, Malaysia
Kok Bin Wong: Institute of Mathematical Sciences, Universiti Malaya, Kuala Lumpur 50603, Malaysia

Mathematics, 2022, vol. 10, issue 2, 1-10

Abstract: By recalling van der Waerden theorem, there exists a least a positive integer w = w ( k ; r ) such that for any n ≥ w , every r -colouring of [ 1 , n ] admits a monochromatic k -term arithmetic progression. Let k ≥ 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 ≤ n ≤ 2 ( k − 1 ) 2 and give an upper bound for r p ( p m ) for any prime p ≥ 3 and integer m ≥ 2 .

Keywords: van der Waerden theorem; monochromatic arithmetic progression (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

Downloads: (external link)
https://www.mdpi.com/2227-7390/10/2/247/pdf (application/pdf)
https://www.mdpi.com/2227-7390/10/2/247/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:10:y:2022:i:2:p:247-:d:724421

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().