The Ramsey property for collections of sequences not containing all arithmetic progressions

Tom C. Brown, Bruce M. Landman

Research output: Contribution to journalArticle

2 Scopus citations

Abstract

A family ℬ of sequences has the Ramsey property if for every positive integer k, there exists a least positive integer f(k) such that for every 2-coloring of {1, 2, . . . , f(k)} there is a monochromatic k-term member of ℬ. For fixed integers m > 1 and 0 ≤ q < m, let ℬq(m) be the collection of those increasing sequences of positive integers {x1 , . . . , xk} such that xi+1 - xi = q(mod m) for 1 ≤ i ≤ k - 1. For t a fixed positive integer, denote by script A signt the collection of those arithmetic progressions having constant difference t. Landman and Long showed that for all m ≥ 2 and 1 ≤ q < m, ℬq(m) does not have the Ramsey property, while ℬq(m) ∪ script A signm does. We extend these results to various finite unions of ℬq(m)'s and script A signt's. We show that for all m ≥ 2, ∪q=1m-1q(m) does not have the Ramsey property. We give necessary and sufficient conditions for collections of the form ℬq(m) ∪ (∪t ∈ T script A signt) to have the Ramsey property. We determine when collections of the form ℬa(m1) ∪ ℬb(m2) have the Ramsey property. We extend this to the study of arbitrary finite unions of ℬq(m)'s. In all cases considered for which ℬ has the Ramsey property, upper bounds are given for f.

Original languageEnglish (US)
Pages (from-to)149-161
Number of pages13
JournalGraphs and Combinatorics
Volume12
Issue number2
StatePublished - Jan 1 1996
Externally publishedYes

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ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics

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