1. On sets without k-term arithmetic progression
- Author
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Shao, Zehui, Deng, Fei, Liang, Meilian, and Xu, Xiaodong
- Subjects
- *
SET theory , *ARITHMETIC series , *NUMBER theory , *INTEGERS , *COMPUTER algorithms , *INFORMATION technology - Abstract
Abstract: For positive integers n and k, let be the size of the largest subset of without arithmetic progressions of length k. The van der Waerden number is the smallest integer w such that every r-coloring of contains a monochromatic -term arithmetic progression with color i for some i. In this note, an algorithm is proposed to search exact values of for some k and n, and some new exact values of for are obtained. The results extend the previous ones significantly. It is also shown that for prime , and three lower bounds for van der Waerden numbers are given: , , . [Copyright &y& Elsevier]
- Published
- 2012
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