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

Pages (from-to) | 19-28 |

Number of pages | 10 |

Journal | Bulletin of the Australian Mathematical Society |

Volume | 55 |

Issue number | 1 |

State | Published - Feb 1 1997 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Bulletin of the Australian Mathematical Society*,

*55*(1), 19-28.

**Collections of sequences having the Ramsey property only for few colours.** / Landman, Bruce M.; Wysocka, Beata.

Research output: Contribution to journal › Article

*Bulletin of the Australian Mathematical Society*, vol. 55, no. 1, pp. 19-28.

}

TY - JOUR

T1 - Collections of sequences having the Ramsey property only for few colours

AU - Landman, Bruce M.

AU - Wysocka, Beata

PY - 1997/2/1

Y1 - 1997/2/1

N2 - 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 {x1, . . . , xk} such that xi - xi-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).

AB - 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 {x1, . . . , xk} such that xi - xi-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).

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M3 - Article

VL - 55

SP - 19

EP - 28

JO - Bulletin of the Australian Mathematical Society

JF - Bulletin of the Australian Mathematical Society

SN - 0004-9727

IS - 1

ER -