On sets without k-term arithmetic progression

  • Authors:

  • Zehui Shao;Fei Deng;Meilian Liang;Xiaodong Xu

  • Affiliations:

  • University Key Laboratory of Pattern Recognition and Intelligent Information Processing Sichuan Province, School of Information Science and Technology, Chengdu University, Chengdu 610106, China;Colleage of Information Science and Technology, Chengdu University of Technology, Chengdu 610059, China;School of Mathematics and Information Science, Guangxi University, Nanning 530004, China;Guangxi Academy of Sciences, Nanning 530007, China

  • Venue:
  • Journal of Computer and System Sciences
  • Year:
  • 2012

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Abstract

For positive integers n and k, let r"k(n) be the size of the largest subset of {1,2,...,n} without arithmetic progressions of length k. The van der Waerden number W(k"1,k"2,...,k"r) is the smallest integer w such that every r-coloring of {1,2,...,w} contains a monochromatic k"i-term arithmetic progression with color i for some i. In this note, an algorithm is proposed to search exact values of r"k(n) for some k and n, and some new exact values of r"k(n) for k=4,5,6,7,8 are obtained. The results extend the previous ones significantly. It is also shown that r"k"+"1(2k^2+1)=2k^2-3k+3 for prime k=3, and three lower bounds for van der Waerden numbers are given: W(3,4,5)=124, W(5,8)=248, W(5,9)=320.