A Set-Coloring Generalization of van der Waerden Numbers
Baoxin Xiu, Guangming Li, Meilian Liang, Xiaodong Xu · Journal of Computational and Theoretical Nanoscience · 2014
Let m > r ≥ 1 and k1 km be integers, and C = color i 1 ≤ i ≤ m be a color set. Let W r k1 km 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 ki 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.