### Abstract

For S ⊆ ℤ^{+} and k and r fixed positive integers, denote by f(S, k; r) the least positive integer n (if it exists) such that within every r-coloring of {1,2,. .., n} there must be a monochromatic sequence {x _{1}, x_{2},....x_{k}} with x_{i} - x _{i-1} ∈ S for 2 ≤ i ≤ k. We consider the existence of f(S, k;r) for various choices of S, as well as upper and lower bounds on this function. In particular, we show that this function exists for all k if 5 is an odd translate of the set of primes and r = 2.

Original language | English (US) |
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Pages (from-to) | 794-801 |

Number of pages | 8 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 21 |

Issue number | 3 |

DOIs | |

State | Published - Dec 1 2007 |

Externally published | Yes |

### Keywords

- Arithmetic progressions
- Primes in progression
- Ramsey theory

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

Landman, B. M., & Robertson, A. (2007). Avoiding monochromatic sequences with special gaps.

*SIAM Journal on Discrete Mathematics*,*21*(3), 794-801. https://doi.org/10.1137/S0895480103422196