Bounds on some van der Waerden numbers

Tom Brown, Bruce M. Landman, Aaron Robertson

Research output: Contribution to journalArticle

7 Scopus citations

Abstract

For positive integers s and k1, k2, ..., ks, the van der Waerden number w (k1, k2, ..., ks ; s) is the minimum integer n such that for every s-coloring of set {1, 2, ..., n}, with colors 1, 2, ..., s, there is a ki-term arithmetic progression of color i for some i. We give an asymptotic lower bound for w (k, m ; 2) for fixed m. We include a table of values of w (k, 3 ; 2) that are very close to this lower bound for m = 3. We also give a lower bound for w (k, k, ..., k ; s) that slightly improves previously-known bounds. Upper bounds for w (k, 4 ; 2) and w (4, 4, ..., 4 ; s) are also provided.

Original languageEnglish (US)
Pages (from-to)1304-1309
Number of pages6
JournalJournal of Combinatorial Theory. Series A
Volume115
Issue number7
DOIs
StatePublished - Oct 1 2008
Externally publishedYes

Keywords

  • Arithmetic progressions
  • Ramsey theory
  • van der Waerden numbers

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics

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