On Generalized Van der Waerden Triples
| dc.creator | Landman, Bruce | |
| dc.creator | Robertson, Aaron | |
| dc.date | 1999-10-19 | |
| dc.date.accessioned | 2026-07-07T05:31:12Z | |
| dc.date.available | 2026-07-07T05:31:12Z | |
| dc.description | Van der Waerden's classical theorem on arithmetic progressions states that for any positive integers k and r, there exists a least positive integer, w(k,r), such that any r-coloring of {1,2,...,w(k,r)} must contain a monochromatic k-term arithmetic progression {x,x+d,x+2d,...,x+(k-1)d}. We investigate the following generalization of w(3,r). For fixed positive integers a and b with a <= b, define N(a,b;r) to be the least positive integer, if it exists, such that any r-coloring of {1,2,...,N(a,b;r)} must contain a monochromatic set of the form {x,ax+d,bx+2d}. We show that N(a,b;2) exists if and only if b <> 2a, and provide upper and lower bounds for it. We then show that for a large class of pairs (a,b), N(a,b;r) does not exist for r sufficiently large. We also give a result on sets of the form {x,ax+d,ax+2d,...,ax+(k-1)d}. | |
| dc.description | 14 pages | |
| dc.identifier | https://arxiv.org/abs/math/9910092 | |
| dc.identifier | http://arxiv.org/abs/math/9910092 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/79255 | |
| dc.subject | Combinatorics | |
| dc.subject | 05D10 | |
| dc.title | On Generalized Van der Waerden Triples | |
| dc.type | text |