On some generalizations of the van der Waerden number w(3)

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2 Scopus citations

Abstract

For integers b≥0 and c≥ 1, define fc(b) to be the least positive integer n such that for every 2-coloring of [1, n] there is a monochromatic sequence of the form {x, x + d, x + 2d + b} where x and d are positive integers with d ≥c. Bialostocki, Lefmann, and Meerdink showed that for b even, 2b + 10≤f⊥(b)≤13/2b + 1, where the lower bound holds for b≥10. We find upper and lower bounds for the more general function fc(b) which, for c = 1, improve the aforementioned upper bound to ⌈9b/4⌉ + 9, and give the same lower bound of 2b + 10. Results about fc(b) are used to find analogous results on a slightly different generalization of f⊥(b).

Original languageEnglish (US)
Pages (from-to)137-147
Number of pages11
JournalDiscrete Mathematics
Volume207
Issue number1-3
DOIs
StatePublished - Sep 28 1999
Externally publishedYes

Keywords

  • Monochromatic sequences
  • Van der Waerden numbers

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics

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