### 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 |

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### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

### Cite this

*Graphs and Combinatorics*,

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

**Ramsey functions for quasi-progressions.** / Landman, Bruce M.

Research output: Contribution to journal › Article

*Graphs and Combinatorics*, vol. 14, no. 2, pp. 131-142. https://doi.org/10.1007/s003730050021

}

TY - JOUR

T1 - Ramsey functions for quasi-progressions

AU - Landman, Bruce M.

PY - 1998/1/1

Y1 - 1998/1/1

N2 - A quasi-progression of diameter n is a finite sequence {x1 , . . . , xk} for which there exists a positive integer L such that L ≤ xi - xi-1 ≤ L + n for i = 2, . . . , k. Let Qn(k) be the least positive integer such that every 2-coloring of {1 , . . . , Qn(k)} will contain a monochromatic k-term quasi-progression of diameter n. We give a lower bound for Qk-i(k) in terms of k and i that holds for all k > i ≥ 1. Upper bounds are obtained for Qn(k) in all cases for which n ≥ [2k/3]. In particular, we show that Q[2k/3](k) ≤ 43/324 k3(1 + o(1)). Exact formulae are found for Qk-1(k) and Qk-2(k). We include a table of computer-generated values of Qn(k), and make several conjectures.

AB - A quasi-progression of diameter n is a finite sequence {x1 , . . . , xk} for which there exists a positive integer L such that L ≤ xi - xi-1 ≤ L + n for i = 2, . . . , k. Let Qn(k) be the least positive integer such that every 2-coloring of {1 , . . . , Qn(k)} will contain a monochromatic k-term quasi-progression of diameter n. We give a lower bound for Qk-i(k) in terms of k and i that holds for all k > i ≥ 1. Upper bounds are obtained for Qn(k) in all cases for which n ≥ [2k/3]. In particular, we show that Q[2k/3](k) ≤ 43/324 k3(1 + o(1)). Exact formulae are found for Qk-1(k) and Qk-2(k). We include a table of computer-generated values of Qn(k), and make several conjectures.

UR - http://www.scopus.com/inward/record.url?scp=10044296697&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=10044296697&partnerID=8YFLogxK

U2 - 10.1007/s003730050021

DO - 10.1007/s003730050021

M3 - Article

AN - SCOPUS:10044296697

VL - 14

SP - 131

EP - 142

JO - Graphs and Combinatorics

JF - Graphs and Combinatorics

SN - 0911-0119

IS - 2

ER -