### Abstract

A quasi-progression of diameter n is a finite sequence {x_{1} , . . . , x_{k}} for which there exists a positive integer L such that L ≤ x_{i} - x_{i-1} ≤ L + n for i = 2, . . . , k. Let Q_{n}(k) be the least positive integer such that every 2-coloring of {1 , . . . , Q_{n}(k)} will contain a monochromatic k-term quasi-progression of diameter n. We give a lower bound for Q_{k-i}(k) in terms of k and i that holds for all k > i ≥ 1. Upper bounds are obtained for Q_{n}(k) in all cases for which n ≥ [2k/3]. In particular, we show that Q_{[2k/3]}(k) ≤ 43/324 k^{3}(1 + o(1)). Exact formulae are found for Q_{k-1}(k) and Q_{k-2}(k). We include a table of computer-generated values of Q_{n}(k), and make several conjectures.

Original language | English (US) |
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Pages (from-to) | 131-142 |

Number of pages | 12 |

Journal | Graphs and Combinatorics |

Volume | 14 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 1998 |

Externally published | Yes |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

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## Cite this

*Graphs and Combinatorics*,

*14*(2), 131-142. https://doi.org/10.1007/s003730050021