On the existence of a reasonable upper bound for the van der Waerden numbers

Raymond N. Greenwell, Bruce M. Landman

Research output: Contribution to journalArticle

9 Scopus citations

Abstract

Numbers similar to those of van der Waerden are examined. We consider increasing sequences of positive integers {x1, x2, ..., xn} either that form an arithmetic sequence or for which there exists a polynomial f(x) = Σi = 0n - 2 aixi with ai ε{lunate} Z, an - 2 > 0, and xj + 1 = f(xj). We denote by q(n) the least positive integer such that if {1, 2, ..., q(n)} is 2-colored, then there exists a monochromatic sequence of the type just described. We give an upper bound for q(n), as well as values of q(n) for n ≤ 5. A stronger upper bound for q(n) is conjectured and is shown to imply the existence of a similar bound on the nth van der Waerden number.

Original languageEnglish (US)
Pages (from-to)82-86
Number of pages5
JournalJournal of Combinatorial Theory, Series A
Volume50
Issue number1
DOIs
StatePublished - Jan 1989
Externally publishedYes

ASJC Scopus subject areas

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

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