### Abstract

A family C of sequences has the r-Ramsey property if for every positive integer k, there exists a least positive integer g^{(r)}(k) such that for every r-colouring of {1,2, . . . ,g(r)(k)} there is a monochromatic k-term member of C. For fixed integers m > 1 and 0 ≤ a < m, define a k-term a (mod m)-sequence to be an increasing sequence of positive integers {x_{1}, . . . , x_{k}} such that x_{i} - x_{i-1} = a (mod m) for i = 2, . . . ,k. Define an m-a.p. to be an arithmetic progression where the difference between successive terms is m. Let C*_{a(m)} be the collection of sequences that are either a (mod m)-sequences or m-a.p.'s. Landman and Long showed that for all m ≥ 2 and 1 ≤ a < m, C*_{a(m)} has the 2-Ramsey property, and that the 2-Ramsey function g^{(2)}_{a(m)}(k, n) , corresponding to k-term a (mod m)-sequences or n-term m-a.p.'s, has order of magnitude mkn. We show that C*_{a(m)} does not have the 4-Ramsey property and that, unless m/a = 2, it does not have the 3-Ramsey property. In the case where m/a = 2, we give an exact formula for g^{(3)}_{a(m)}(k,n). We show that if a ≠ 0, there exist 4-colourings or 6-colourings (depending on m and a) of the positive integers which avoid 2-term monochromatic members of C*_{a(m)}, but that there never exist such 3-colourings. We also give an exact formula for g^{(r)}_{0(m)}(k, n).

Original language | English (US) |
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Pages (from-to) | 19-28 |

Number of pages | 10 |

Journal | Bulletin of the Australian Mathematical Society |

Volume | 55 |

Issue number | 1 |

DOIs | |

State | Published - Feb 1997 |

Externally published | Yes |

### ASJC Scopus subject areas

- Mathematics(all)