Hostname: page-component-76fb5796d-x4r87 Total loading time: 0 Render date: 2024-04-27T07:21:08.649Z Has data issue: false hasContentIssue false
  • English
  • Français

A Construction for Partitions Which Avoid Long Arithmetic Progressions

Published online by Cambridge University Press:  20 November 2018

E.R. Berlekamp
E.R. Berlekamp*
Affiliation:
Bell Telephone Laboratories Incorporated, Murray Hill, New Jersey
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For k ≥2, t ≥2, let W(k, t) denote the least integer m such that in every partition of m consecutive integers into k sets, atleast one set contains an arithmetic progression of t+1 terms. This paper presents a construction which improves the best previously known lower bounds on W(k, t) for small k and large t.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 11 , Issue 3 , August 1968 , pp. 409 - 414
Copyright
Copyright © Canadian Mathematical Society 1968

References

Erdös, P. and Radó, R., Combinatorial theorems on classifications of subsets of a given set. Proceedings of the London Mathematical Society (3) 2(1952)417-439.Google Scholar
Folkman, J., private communication (1967).Google Scholar
Moser, L., On a theorem of van der Waerden. Canadian Mathematical Bulletin, 3 (1960) 23-25.Google Scholar
Schmidt, W.M., Two combinatorial theorems on arithmetic progressions. Duke Math. J. 29(1962)129-140.Google Scholar
van der Waerden, B.L., Beweis einer Baudet'schen Vermutung. Niew Archief voor Wiskunde, 15 (1925) 212-216.Google Scholar