### Abstract

A family F of sequences has the r-Ramsey property if for every positive integer k, there exists a least positive integer f = f^{(r)}(F,k) such that for every r-coloring of {1,2,...,f} there is a monochromatic k-term member of F. For fixed integers 1 ≤ a < m, a k-term a(mod m)-sequence is an increasing sequence of positive integers {x_{1},... ,x_{k}} such that x_{i}-x_{i} ≡ a(mod m) for i = 2,... ,k. An arithmetic progression modulo m is a sequence that is an a(mod m)-sequence for some a ∈ {1,2,... , m - 1}. A d-a.p. is an arithmetic progression where the difference between successive terms is d. It is known that if 1 ≤ a < m are fixed, then the family consisting of all a(mod m)-sequences and all m-a.p.'s has the 2-Ramsey property, does not have the 4-Ramsey property, and has the 3-Ramsey property only when a = m/2. This paper extends these results to much larger families. In particular, we show that if D contains at most a finite number of multiples of m, then the family of sequences that are either arithmetic progressions modulo m, or are d-a.p.'s for some d ∈ D, does not have the 4-Ramsey property, and has the 3-Ramsey property if and only if m is even and D contains some multiple of m.

Original language | English (US) |
---|---|

Pages (from-to) | 173-182 |

Number of pages | 10 |

Journal | Utilitas Mathematica |

Volume | 52 |

State | Published - Dec 1 1997 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty
- Applied Mathematics

### Cite this

**Avoiding arithmetic progressions (mod m) and arithmetic progressions.** / Landman, Bruce M.

Research output: Contribution to journal › Article

*Utilitas Mathematica*, vol. 52, pp. 173-182.

}

TY - JOUR

T1 - Avoiding arithmetic progressions (mod m) and arithmetic progressions

AU - Landman, Bruce M.

PY - 1997/12/1

Y1 - 1997/12/1

N2 - A family F of sequences has the r-Ramsey property if for every positive integer k, there exists a least positive integer f = f(r)(F,k) such that for every r-coloring of {1,2,...,f} there is a monochromatic k-term member of F. For fixed integers 1 ≤ a < m, a k-term a(mod m)-sequence is an increasing sequence of positive integers {x1,... ,xk} such that xi-xi ≡ a(mod m) for i = 2,... ,k. An arithmetic progression modulo m is a sequence that is an a(mod m)-sequence for some a ∈ {1,2,... , m - 1}. A d-a.p. is an arithmetic progression where the difference between successive terms is d. It is known that if 1 ≤ a < m are fixed, then the family consisting of all a(mod m)-sequences and all m-a.p.'s has the 2-Ramsey property, does not have the 4-Ramsey property, and has the 3-Ramsey property only when a = m/2. This paper extends these results to much larger families. In particular, we show that if D contains at most a finite number of multiples of m, then the family of sequences that are either arithmetic progressions modulo m, or are d-a.p.'s for some d ∈ D, does not have the 4-Ramsey property, and has the 3-Ramsey property if and only if m is even and D contains some multiple of m.

AB - A family F of sequences has the r-Ramsey property if for every positive integer k, there exists a least positive integer f = f(r)(F,k) such that for every r-coloring of {1,2,...,f} there is a monochromatic k-term member of F. For fixed integers 1 ≤ a < m, a k-term a(mod m)-sequence is an increasing sequence of positive integers {x1,... ,xk} such that xi-xi ≡ a(mod m) for i = 2,... ,k. An arithmetic progression modulo m is a sequence that is an a(mod m)-sequence for some a ∈ {1,2,... , m - 1}. A d-a.p. is an arithmetic progression where the difference between successive terms is d. It is known that if 1 ≤ a < m are fixed, then the family consisting of all a(mod m)-sequences and all m-a.p.'s has the 2-Ramsey property, does not have the 4-Ramsey property, and has the 3-Ramsey property only when a = m/2. This paper extends these results to much larger families. In particular, we show that if D contains at most a finite number of multiples of m, then the family of sequences that are either arithmetic progressions modulo m, or are d-a.p.'s for some d ∈ D, does not have the 4-Ramsey property, and has the 3-Ramsey property if and only if m is even and D contains some multiple of m.

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M3 - Article

VL - 52

SP - 173

EP - 182

JO - Utilitas Mathematica

JF - Utilitas Mathematica

SN - 0315-3681

ER -