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 language | English (US) |
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Pages (from-to) | 1304-1309 |
Number of pages | 6 |
Journal | Journal of Combinatorial Theory. Series A |
Volume | 115 |
Issue number | 7 |
DOIs | |
State | Published - Oct 2008 |
Externally published | Yes |
Keywords
- Arithmetic progressions
- Ramsey theory
- van der Waerden numbers
ASJC Scopus subject areas
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics