A Set-Coloring Generalization of van der Waerden Numbers

作者：Xiu, Baoxin; Li, Guangming; Liang, Meilian^*; Xu, Xiaodong

来源：Journal of Computational and Theoretical Nanoscience, 2014, 11(12): 2431-2436.

DOI：10.1166/jctn.2014.3658

• Arithmetic Progression
• van der Waerden Number
• Set-Coloring
• Szemeredi's Theorem
• Algorithm

Let m > r >= 1 and k(1), ... , k(m) be integers, and C = {color i vertical bar <= i <= m} be a color set. Let W-(r) (k(1), ... , k(m)) be the smallest positive integer n such that if any integer in {1, ... , n} is colored with an r-subset of C, then there must exist an arithmetic progression of k(i) terms in which any integer is colored with an r-subset of C containing color i. In this paper, such a set-coloring generalization of the van der Waerden number is proposed and studied. By studying such a set-coloring generalization, van der Waerden numbers can be understood better. On the other hand, computing set-coloring van der Waerden numbers can be regarded as a new challenge.

• 出版日期2014-12
• 单位广西大学; 中国人民解放军国防科学技术大学; 广西科学院