The Ramsey property for collections of sequences not containing all arithmetic progressions |
| |
Authors: | Tom C Brown Bruce M Landman |
| |
Institution: | (1) Department of Mathematics and Statistics, Simon Fraser University, V5A 1S6 Burnaby, British Columbia, Canada;(2) Department of Mathematical Sciences, University of North Carolina at Greensboro, 27412 North Carolina, USA |
| |
Abstract: | A family of sequences has the Ramsey property if for every positive integerk, there exists a least positive integerf
(k) such that for every 2-coloring of {1,2, ...,f
(k)} there is a monochromatick-term member of . For fixed integersm > 1 and 0 q < m, let q(m) be the collection of those increasing sequences of positive integers {x
1,..., xk} such thatx
i+1 – xi q(modm) for 1 i k – 1. Fort a fixed positive integer, denote byA
t
the collection of those arithmetic progressions having constant differencet. Landman and Long showed that for allm 2 and 1 q < m,
q(m) does not have the Ramsey property, while
q(m)
A
m
does. We extend these results to various finite unions of
q(m)
's andA
t
's. We show that for allm 2,
q=1
m–1
q(m) does not have the Ramsey property. We give necessary and sufficient conditions for collections of the form
q(m)
(
t T
A
t) to have the Ramsey property. We determine when collections of the form a(m1) b(m2) have the Ramsey property. We extend this to the study of arbitrary finite unions of q(m)'s. In all cases considered for which has the Ramsey property, upper bounds are given forf
. |
| |
Keywords: | |
本文献已被 SpringerLink 等数据库收录! |
|