Avoiding monochromatic sequences with special gaps

Bruce M. Landman, Aaron Robertson

Research output: Contribution to journalArticle

5 Scopus citations

Abstract

For S ⊆ ℤ+ and k and r fixed positive integers, denote by f(S, k; r) the least positive integer n (if it exists) such that within every r-coloring of {1,2,. .., n} there must be a monochromatic sequence {x 1, x2,....xk} with xi - x i-1 ∈ S for 2 ≤ i ≤ k. We consider the existence of f(S, k;r) for various choices of S, as well as upper and lower bounds on this function. In particular, we show that this function exists for all k if 5 is an odd translate of the set of primes and r = 2.

Original languageEnglish (US)
Pages (from-to)794-801
Number of pages8
JournalSIAM Journal on Discrete Mathematics
Volume21
Issue number3
DOIs
StatePublished - Dec 1 2007
Externally publishedYes

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Keywords

  • Arithmetic progressions
  • Primes in progression
  • Ramsey theory

ASJC Scopus subject areas

  • Mathematics(all)

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