### Abstract

For a positive integer d, a set S of positive integers is difference d-tree if \x-y\ # d for all x, y ε S. We consider the following Ramseytheoretical question: Given d, k, r ε Z^{+}, what is the smallest integer n such that every r-coloring of [1, n] contains a monochromatic k-element difference d-free set? We provide a formula for this n. We then consider the more general problem where the monochromatic fc-element set must avoid a given set of differences rather than just one difference.

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

Pages (from-to) | 11-20 |

Number of pages | 10 |

Journal | Journal of Combinatorial Mathematics and Combinatorial Computing |

Volume | 76 |

State | Published - Feb 1 2011 |

### Fingerprint

### Keywords

- Difference-free sets
- Integer ramsey theory
- Monochromatic sets

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Journal of Combinatorial Mathematics and Combinatorial Computing*,

*76*, 11-20.

**On Ramsey numbers for sets free of prescribed differences.** / Landman, Bruce M.; Perconti, James T.

Research output: Contribution to journal › Article

*Journal of Combinatorial Mathematics and Combinatorial Computing*, vol. 76, pp. 11-20.

}

TY - JOUR

T1 - On Ramsey numbers for sets free of prescribed differences

AU - Landman, Bruce M.

AU - Perconti, James T.

PY - 2011/2/1

Y1 - 2011/2/1

N2 - For a positive integer d, a set S of positive integers is difference d-tree if \x-y\ # d for all x, y ε S. We consider the following Ramseytheoretical question: Given d, k, r ε Z+, what is the smallest integer n such that every r-coloring of [1, n] contains a monochromatic k-element difference d-free set? We provide a formula for this n. We then consider the more general problem where the monochromatic fc-element set must avoid a given set of differences rather than just one difference.

AB - For a positive integer d, a set S of positive integers is difference d-tree if \x-y\ # d for all x, y ε S. We consider the following Ramseytheoretical question: Given d, k, r ε Z+, what is the smallest integer n such that every r-coloring of [1, n] contains a monochromatic k-element difference d-free set? We provide a formula for this n. We then consider the more general problem where the monochromatic fc-element set must avoid a given set of differences rather than just one difference.

KW - Difference-free sets

KW - Integer ramsey theory

KW - Monochromatic sets

UR - http://www.scopus.com/inward/record.url?scp=79952568630&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79952568630&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:79952568630

VL - 76

SP - 11

EP - 20

JO - Journal of Combinatorial Mathematics and Combinatorial Computing

JF - Journal of Combinatorial Mathematics and Combinatorial Computing

SN - 0835-3026

ER -