Bounds on some van der Waerden numbers

Tom Brown, Bruce M. Landman, Aaron Robertson

Research output: Contribution to journalArticle

7 Citations (Scopus)

Abstract

For positive integers s and k1, k2, ..., ks, the van der Waerden number w (k1, k2, ..., ks ; s) is the minimum integer n such that for every s-coloring of set {1, 2, ..., n}, with colors 1, 2, ..., s, there is a ki-term arithmetic progression of color i for some i. We give an asymptotic lower bound for w (k, m ; 2) for fixed m. We include a table of values of w (k, 3 ; 2) that are very close to this lower bound for m = 3. We also give a lower bound for w (k, k, ..., k ; s) that slightly improves previously-known bounds. Upper bounds for w (k, 4 ; 2) and w (4, 4, ..., 4 ; s) are also provided.

Original languageEnglish (US)
Pages (from-to)1304-1309
Number of pages6
JournalJournal of Combinatorial Theory. Series A
Volume115
Issue number7
DOIs
StatePublished - Oct 1 2008

Fingerprint

Lower bound
Color
Coloring
Integer
Arithmetic sequence
Colouring
Table
Upper bound
Term

Keywords

  • Arithmetic progressions
  • Ramsey theory
  • van der Waerden numbers

ASJC Scopus subject areas

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

Cite this

Bounds on some van der Waerden numbers. / Brown, Tom; Landman, Bruce M.; Robertson, Aaron.

In: Journal of Combinatorial Theory. Series A, Vol. 115, No. 7, 01.10.2008, p. 1304-1309.

Research output: Contribution to journalArticle

Brown, Tom ; Landman, Bruce M. ; Robertson, Aaron. / Bounds on some van der Waerden numbers. In: Journal of Combinatorial Theory. Series A. 2008 ; Vol. 115, No. 7. pp. 1304-1309.
@article{25c0b5b926a0402baed42647bdd89d7d,
title = "Bounds on some van der Waerden numbers",
abstract = "For positive integers s and k1, k2, ..., ks, the van der Waerden number w (k1, k2, ..., ks ; s) is the minimum integer n such that for every s-coloring of set {1, 2, ..., n}, with colors 1, 2, ..., s, there is a ki-term arithmetic progression of color i for some i. We give an asymptotic lower bound for w (k, m ; 2) for fixed m. We include a table of values of w (k, 3 ; 2) that are very close to this lower bound for m = 3. We also give a lower bound for w (k, k, ..., k ; s) that slightly improves previously-known bounds. Upper bounds for w (k, 4 ; 2) and w (4, 4, ..., 4 ; s) are also provided.",
keywords = "Arithmetic progressions, Ramsey theory, van der Waerden numbers",
author = "Tom Brown and Landman, {Bruce M.} and Aaron Robertson",
year = "2008",
month = "10",
day = "1",
doi = "10.1016/j.jcta.2008.01.005",
language = "English (US)",
volume = "115",
pages = "1304--1309",
journal = "Journal of Combinatorial Theory - Series A",
issn = "0097-3165",
publisher = "Academic Press Inc.",
number = "7",

}

TY - JOUR

T1 - Bounds on some van der Waerden numbers

AU - Brown, Tom

AU - Landman, Bruce M.

AU - Robertson, Aaron

PY - 2008/10/1

Y1 - 2008/10/1

N2 - For positive integers s and k1, k2, ..., ks, the van der Waerden number w (k1, k2, ..., ks ; s) is the minimum integer n such that for every s-coloring of set {1, 2, ..., n}, with colors 1, 2, ..., s, there is a ki-term arithmetic progression of color i for some i. We give an asymptotic lower bound for w (k, m ; 2) for fixed m. We include a table of values of w (k, 3 ; 2) that are very close to this lower bound for m = 3. We also give a lower bound for w (k, k, ..., k ; s) that slightly improves previously-known bounds. Upper bounds for w (k, 4 ; 2) and w (4, 4, ..., 4 ; s) are also provided.

AB - For positive integers s and k1, k2, ..., ks, the van der Waerden number w (k1, k2, ..., ks ; s) is the minimum integer n such that for every s-coloring of set {1, 2, ..., n}, with colors 1, 2, ..., s, there is a ki-term arithmetic progression of color i for some i. We give an asymptotic lower bound for w (k, m ; 2) for fixed m. We include a table of values of w (k, 3 ; 2) that are very close to this lower bound for m = 3. We also give a lower bound for w (k, k, ..., k ; s) that slightly improves previously-known bounds. Upper bounds for w (k, 4 ; 2) and w (4, 4, ..., 4 ; s) are also provided.

KW - Arithmetic progressions

KW - Ramsey theory

KW - van der Waerden numbers

UR - http://www.scopus.com/inward/record.url?scp=50649125454&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=50649125454&partnerID=8YFLogxK

U2 - 10.1016/j.jcta.2008.01.005

DO - 10.1016/j.jcta.2008.01.005

M3 - Article

VL - 115

SP - 1304

EP - 1309

JO - Journal of Combinatorial Theory - Series A

JF - Journal of Combinatorial Theory - Series A

SN - 0097-3165

IS - 7

ER -