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

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 |

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### Keywords

- Arithmetic progressions
- Primes in progression
- Ramsey theory

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*SIAM Journal on Discrete Mathematics*,

*21*(3), 794-801. https://doi.org/10.1137/S0895480103422196

**Avoiding monochromatic sequences with special gaps.** / Landman, Bruce M.; Robertson, Aaron.

Research output: Contribution to journal › Article

*SIAM Journal on Discrete Mathematics*, vol. 21, no. 3, pp. 794-801. https://doi.org/10.1137/S0895480103422196

}

TY - JOUR

T1 - Avoiding monochromatic sequences with special gaps

AU - Landman, Bruce M.

AU - Robertson, Aaron

PY - 2007/12/1

Y1 - 2007/12/1

N2 - 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, x2,....xk} with xi - 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.

AB - 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, x2,....xk} with xi - 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.

KW - Arithmetic progressions

KW - Primes in progression

KW - Ramsey theory

UR - http://www.scopus.com/inward/record.url?scp=49449102333&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=49449102333&partnerID=8YFLogxK

U2 - 10.1137/S0895480103422196

DO - 10.1137/S0895480103422196

M3 - Article

AN - SCOPUS:49449102333

VL - 21

SP - 794

EP - 801

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

SN - 0895-4801

IS - 3

ER -