## Abstract

In 1977 L.T. Ramsey showed that any sequence in Z^{2} with bounded gaps contains arbitrarily many collinear points. Thereafter, in 1980, C. Pomerance provided a density version of this result, relaxing the condition on the sequence from having bounded gaps to having gaps bounded on average.We give a higher dimensional generalization of these results. Our main theorem is the following. Theorem. Let d∈N, let f:Z^{d}→Z^{d+1} be a Lipschitz map and let A⊂Z^{d} have positive upper Banach density. Then f(A) contains arbitrarily many collinear points.Note that Pomerance's theorem corresponds to the special case d= 1. In our proof, we transfer the problem from a discrete to a continuous setting, allowing us to take advantage of analytic and measure theoretic tools such as Rademacher's theorem.

Original language | English (US) |
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Pages (from-to) | 163-176 |

Number of pages | 14 |

Journal | European Journal of Combinatorics |

Volume | 54 |

DOIs | |

State | Published - May 1 2016 |

## ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics

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