### Abstract

A sequence of integers x_{1} < x_{2} < ··· < x_{k} is called an ascending wave of length k if x_{i + 1} - x_{i} ≤ x_{i + 2} - x_{i + 1} for all 1 ≤ i ≤ k - 2. Let f(k) be the smallest positive integer such that any 2-coloring of {1, 2, ..., f(k)} contains a monochromatic ascending wave of length k. Settling a problem of Brown, Erdös, and Freedman we show that there are two positive constants c_{1}, c_{2} such that c_{1}k^{3} ≤ f(k) ≤ c_{2}k^{3} for all k ≥ 1. Let g(n) be the largest integer k such that any set A ⊆ {1, 2, ..., n} of cardinality |A| ≥ n 2 contains an ascending wave of length k. We show that there are two positive constants c_{3} and c_{4} such that c_{3}(log n)^{2} log log n ≤ g(n) ≤ c_{4}(log n)^{2} for all n ≥ 1.

Original language | English (US) |
---|---|

Pages (from-to) | 275-287 |

Number of pages | 13 |

Journal | Journal of Combinatorial Theory, Series A |

Volume | 52 |

Issue number | 2 |

DOIs | |

State | Published - 1989 |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Journal of Combinatorial Theory, Series A*,

*52*(2), 275-287. https://doi.org/10.1016/0097-3165(89)90033-2

**Ascending waves.** / Alon, N.; Spencer, Joel.

Research output: Contribution to journal › Article

*Journal of Combinatorial Theory, Series A*, vol. 52, no. 2, pp. 275-287. https://doi.org/10.1016/0097-3165(89)90033-2

}

TY - JOUR

T1 - Ascending waves

AU - Alon, N.

AU - Spencer, Joel

PY - 1989

Y1 - 1989

N2 - A sequence of integers x1 < x2 < ··· < xk is called an ascending wave of length k if xi + 1 - xi ≤ xi + 2 - xi + 1 for all 1 ≤ i ≤ k - 2. Let f(k) be the smallest positive integer such that any 2-coloring of {1, 2, ..., f(k)} contains a monochromatic ascending wave of length k. Settling a problem of Brown, Erdös, and Freedman we show that there are two positive constants c1, c2 such that c1k3 ≤ f(k) ≤ c2k3 for all k ≥ 1. Let g(n) be the largest integer k such that any set A ⊆ {1, 2, ..., n} of cardinality |A| ≥ n 2 contains an ascending wave of length k. We show that there are two positive constants c3 and c4 such that c3(log n)2 log log n ≤ g(n) ≤ c4(log n)2 for all n ≥ 1.

AB - A sequence of integers x1 < x2 < ··· < xk is called an ascending wave of length k if xi + 1 - xi ≤ xi + 2 - xi + 1 for all 1 ≤ i ≤ k - 2. Let f(k) be the smallest positive integer such that any 2-coloring of {1, 2, ..., f(k)} contains a monochromatic ascending wave of length k. Settling a problem of Brown, Erdös, and Freedman we show that there are two positive constants c1, c2 such that c1k3 ≤ f(k) ≤ c2k3 for all k ≥ 1. Let g(n) be the largest integer k such that any set A ⊆ {1, 2, ..., n} of cardinality |A| ≥ n 2 contains an ascending wave of length k. We show that there are two positive constants c3 and c4 such that c3(log n)2 log log n ≤ g(n) ≤ c4(log n)2 for all n ≥ 1.

UR - http://www.scopus.com/inward/record.url?scp=38249004682&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=38249004682&partnerID=8YFLogxK

U2 - 10.1016/0097-3165(89)90033-2

DO - 10.1016/0097-3165(89)90033-2

M3 - Article

AN - SCOPUS:38249004682

VL - 52

SP - 275

EP - 287

JO - Journal of Combinatorial Theory - Series A

JF - Journal of Combinatorial Theory - Series A

SN - 0097-3165

IS - 2

ER -