A Radix Representation for each van der Waerden number W(r, k) with r colors: Why log_rW(r, k) < k² is true whenever k is the number of terms in the arithmetic progression

2016/04/24 by Robert J Betts, Betts, Robert J
Computer Science · #11A63 #11B25 (Primary) #68R01 (Secondary) #Discrete Mathematics (cs.DM) #F.2.1 #FOS: Computer and information sciences #G.2.0 #acm:11A63 #acm:11B25 #acm:68R01 #cs.DM #msc:11A63 #msc:11B25 #msc:68R01

paper · pdf · doi:10.48550/arxiv.1604.07036

Seven pages, two tables, no figures. Two sentences added to Section 1, paragraph two. Additional section added (Section 2) to derive a mathematical expression for the rational number $\frac{W(r, k + 1)}{W(r, k)}$

arxiv created 2016/05/07 · arxiv updated 2016/05/10

Abstract

Here we show that by expressing a van der Waerden number W(r, k) by its radix polynomial representation, it not only is possible to locate each proper subset on ℝ in which the van der Waerden number lies, but also to show that conditions exist for which the logarithm of the van der Waerden number necessarily is bounded above by the square of the number of terms k in the arithmetic progression. Furthermore we also use the method to find a mathematical expression or formula for the ratio of two "consecutive" van der Waerden numbers of the kind W(r, k), W(r, k + 1).

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