### Abstract

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

Original language | English (US) |
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Pages (from-to) | 149-161 |

Number of pages | 13 |

Journal | Graphs and Combinatorics |

Volume | 12 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 1996 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics