### 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) |
---|---|

Pages (from-to) | 149-161 |

Number of pages | 13 |

Journal | Graphs and Combinatorics |

Volume | 12 |

Issue number | 1 |

State | Published - Dec 1 1996 |

### Fingerprint

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

### Cite this

*Graphs and Combinatorics*,

*12*(1), 149-161.

**The Ramsey property for collections of sequences not containing all arithmetic progressions.** / Brown, Tom C.; Landman, Bruce M.

Research output: Contribution to journal › Article

*Graphs and Combinatorics*, vol. 12, no. 1, pp. 149-161.

}

TY - JOUR

T1 - The Ramsey property for collections of sequences not containing all arithmetic progressions

AU - Brown, Tom C.

AU - Landman, Bruce M.

PY - 1996/12/1

Y1 - 1996/12/1

N2 - A family ℬ of sequences has the Ramsey property if for every positive integer k, there exists a least positive integer fℬ(k) such that for every 2-coloring of {1, 2, . . . , fℬ(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-1 ℬq(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 fℬ.

AB - A family ℬ of sequences has the Ramsey property if for every positive integer k, there exists a least positive integer fℬ(k) such that for every 2-coloring of {1, 2, . . . , fℬ(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-1 ℬq(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 fℬ.

UR - http://www.scopus.com/inward/record.url?scp=10044253858&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=10044253858&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:10044253858

VL - 12

SP - 149

EP - 161

JO - Graphs and Combinatorics

JF - Graphs and Combinatorics

SN - 0911-0119

IS - 1

ER -