2015/12/11 by Robert J. Betts, Robert J Betts, Betts, Robert J
Computer Science · Engineering · Mathematics · #11P99 (Primary) #68R01 (Secondary) #Advanced Graph Theory Research #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #G.2.0 #Limits and Structures in Graph Theory #acm:11P99 #acm:68R01 #cs.DM #graph theory and CDMA systems #msc:11P99 #msc:68R01 #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1512.03631
25 pages, one Table, no figures. Small addition to Section 1. Slight revision of Theorem 3.1, Section 2. Typo correction for Theorem 7.2, Section 7, where \(\frac{k}{r} = o(1)\) ought to have been \(\frac{k}{r^{n}} = o(1)\) and \(k \ll r^{n}\)
openalex publication_date 2015/12/11 · arxiv created 2016/01/26 · arxiv updated 2016/01/27 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
What is a least integer upper bound on van der Waerden number W(r, k) among the powers of the integer r? We show how this can be found by expanding the integer W(r, k) into powers of r. Doing this enables us to find both a least upper bound and a greatest lower bound on W(r, k) that are some powers of r and where the greatest lower bound is equal to or smaller than W(r, k). A finite series expansion of each W(r, k) into integer powers of r then helps us to find also a greatest real lower bound on any k for which a conjecture posed by R. Graham is true, following immediately as a particular case of the overall result.