### Abstract

For each positive integer n, G(n) is defined to be the largest integer k such that no matter how Z_{n} is two-colored, some progression a, a + d, a + 2d, ..., a + (k - 1)d of k distinct elements of Z_{n} will appear in one color. Our main theorem shows constructively that if Z_{n} can be two-colored in such a way that the longest monochrome progression has m distinct terms mod n and any monochrome progression of k distinct terms with common difference d ≢ 0 (mod n) has the property that kd ≢ 0 (mod n), then G(rn) ≤ m for 1 ≤ r ≤ m and G(rn) ≤ r for r > m. One lower bound on G(n) says G(rn) ≥ G(n) for r ≥ 1 and n ≥ 1. Two main results with corollaries, a "quadratic-residue coloration" on Z_{p} for p a prime, and the van der Waerden numbers W(k) for 2 ≤ k ≤ 5, together with a computer search, have been used to determine the exact value of G(n) for 1 ≤ n ≤ 53, for all primes up to 71, and for a few more cases, the highest of which is G(695) = 5 which guarantees that W(6) ≥ 696.

Original language | English |
---|---|

Pages (from-to) | 211-221 |

Number of pages | 11 |

Journal | Journal of Combinatorial Theory, Series A |

Volume | 61 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1992 Nov |

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### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

### Cite this

_{n}.

*Journal of Combinatorial Theory, Series A*,

*61*(2), 211-221. https://doi.org/10.1016/0097-3165(92)90018-P

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_{n}',

*Journal of Combinatorial Theory, Series A*, vol. 61, no. 2, pp. 211-221. https://doi.org/10.1016/0097-3165(92)90018-P

**Progressions in every two-coloration of Z _{n}.** / Song, H. Y.; Taylor, H.; Golomb, S. W.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Progressions in every two-coloration of Zn

AU - Song, H. Y.

AU - Taylor, H.

AU - Golomb, S. W.

PY - 1992/11

Y1 - 1992/11

N2 - For each positive integer n, G(n) is defined to be the largest integer k such that no matter how Zn is two-colored, some progression a, a + d, a + 2d, ..., a + (k - 1)d of k distinct elements of Zn will appear in one color. Our main theorem shows constructively that if Zn can be two-colored in such a way that the longest monochrome progression has m distinct terms mod n and any monochrome progression of k distinct terms with common difference d ≢ 0 (mod n) has the property that kd ≢ 0 (mod n), then G(rn) ≤ m for 1 ≤ r ≤ m and G(rn) ≤ r for r > m. One lower bound on G(n) says G(rn) ≥ G(n) for r ≥ 1 and n ≥ 1. Two main results with corollaries, a "quadratic-residue coloration" on Zp for p a prime, and the van der Waerden numbers W(k) for 2 ≤ k ≤ 5, together with a computer search, have been used to determine the exact value of G(n) for 1 ≤ n ≤ 53, for all primes up to 71, and for a few more cases, the highest of which is G(695) = 5 which guarantees that W(6) ≥ 696.

AB - For each positive integer n, G(n) is defined to be the largest integer k such that no matter how Zn is two-colored, some progression a, a + d, a + 2d, ..., a + (k - 1)d of k distinct elements of Zn will appear in one color. Our main theorem shows constructively that if Zn can be two-colored in such a way that the longest monochrome progression has m distinct terms mod n and any monochrome progression of k distinct terms with common difference d ≢ 0 (mod n) has the property that kd ≢ 0 (mod n), then G(rn) ≤ m for 1 ≤ r ≤ m and G(rn) ≤ r for r > m. One lower bound on G(n) says G(rn) ≥ G(n) for r ≥ 1 and n ≥ 1. Two main results with corollaries, a "quadratic-residue coloration" on Zp for p a prime, and the van der Waerden numbers W(k) for 2 ≤ k ≤ 5, together with a computer search, have been used to determine the exact value of G(n) for 1 ≤ n ≤ 53, for all primes up to 71, and for a few more cases, the highest of which is G(695) = 5 which guarantees that W(6) ≥ 696.

UR - http://www.scopus.com/inward/record.url?scp=38249009267&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=38249009267&partnerID=8YFLogxK

U2 - 10.1016/0097-3165(92)90018-P

DO - 10.1016/0097-3165(92)90018-P

M3 - Article

AN - SCOPUS:38249009267

VL - 61

SP - 211

EP - 221

JO - Journal of Combinatorial Theory - Series A

JF - Journal of Combinatorial Theory - Series A

SN - 0097-3165

IS - 2

ER -

_{n}. Journal of Combinatorial Theory, Series A. 1992 Nov;61(2):211-221. https://doi.org/10.1016/0097-3165(92)90018-P