### Abstract

Let m ≥ 3 and k ≥ 1 be two given integers. A sub-k-coloring of [n] = {1, 2,..., n} is an assignment of colors to the numbers of [n] in which each color is used at most k times. Call an {Mathematical expression} a rainbow set if no two of its elements have the same color. The sub-k-Ramsey number sr(m, k) is defined as the minimum n such that every sub-k-coloring of [n] contains a rainbow arithmetic progression of m terms. We prove that Ω((k - 1)m^{2}/log mk) ≤ sr(m, k) ≤ O((k - 1)m^{2} log mk) as m → ∞, and apply the same method to improve a previously known upper bound for a problem concerning mappings from [n] to [n] without fixed points.

Original language | English (US) |
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Pages (from-to) | 307-314 |

Number of pages | 8 |

Journal | Graphs and Combinatorics |

Volume | 5 |

Issue number | 1 |

DOIs | |

State | Published - Dec 1 1989 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

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## Cite this

Alon, N., Caro, Y., & Tuza, Z. (1989). Sub-Ramsey numbers for arithmetic progressions.

*Graphs and Combinatorics*,*5*(1), 307-314. https://doi.org/10.1007/BF01788685