Some new van der Waerden numbers and some van der Waerden-type numbers
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Tanbir Ahmed
Abstract
The van der Waerden number w(r; k1, k2, . . . , kr) is the least m such that given any partition {1, 2, . . . , m} = P1 ∪ P2 ∪ ⋯ ∪ Pr, there is an index j ∈ {1, 2, . . . , r} such that Pj contains an arithmetic progression of length kj. We have computed exact values of some (30) previously unknown van der Waerden numbers and also computed lower bounds of others. Let wd(r; k1, k2, . . . , kr) be the least m such that given any partition {1, 2, . . . , m} = P1 ∪ P2 ∪ ⋯ ∪ Pr, there is an index j ∈ {1, 2, . . . , r – 1} such that Pj contains an arithmetic progression of length kj, or Pr contains an arithmetic progression of length kr with common difference at most d. A table of observed values of wd(r; k1, k2, . . . , kr) for d = 1, 2, . . . , 
is given.
© de Gruyter 2009
Articles in the same Issue
- On k-Imperfect Numbers
- On Universal Binary Hermitian Forms
- The Shortest Game of Chinese Checkers and Related Problems
- On Two-Point Configurations in a Random Set
- On Rapid Generation of SL2(𝔽q)
- Tiling Proofs of Some Formulas for the Pell Numbers of Odd Index
- Some new van der Waerden numbers and some van der Waerden-type numbers
- Integer Partitions into Arithmetic Progressions with an Odd Common Difference
- On Asymptotic Constants Related to Products of Bernoulli Numbers and Factorials
Articles in the same Issue
- On k-Imperfect Numbers
- On Universal Binary Hermitian Forms
- The Shortest Game of Chinese Checkers and Related Problems
- On Two-Point Configurations in a Random Set
- On Rapid Generation of SL2(𝔽q)
- Tiling Proofs of Some Formulas for the Pell Numbers of Odd Index
- Some new van der Waerden numbers and some van der Waerden-type numbers
- Integer Partitions into Arithmetic Progressions with an Odd Common Difference
- On Asymptotic Constants Related to Products of Bernoulli Numbers and Factorials
