Distinguishing subgroups of the rationals by their Ramsey properties

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:129

Article

Barber, Ben¹  Hindman, Neil²  Leader, Imre³  Strauss, Dona⁴ 

[1] Univ Birmingham, Sch Math, Birmingham B15 2TT, W Midlands, England

[2] Howard Univ, Dept Math, Washington, DC 20059 USA

[3] Ctr Math Sci, Dept Pure Math & Math Stat, Cambridge CB3 0WB, England

[4] Univ Leeds, Dept Pure Math, Leeds LS2 9J2, W Yorkshire, England

关键词: Partion regular;    Rationals;    Subgroups;    Central sets;   

DOI  :  10.1016/j.jcta.2014.10.002

来源: Elsevier

【 摘 要 】

A system of linear equations with integer coefficients is partition regular over a subset S of the reals if, whenever S \ {0} is finitely coloured, there is a solution to the system contained in one colour class. It has been known for some time that there is an infinite system of linear equations that is partition regular over R but not over Q, and it was recently shown (answering a long-standing open question) that one can also distinguish Q from Z in this way. Our aim is to show that the transition from Z to Q is not sharp: there is an infinite chain of subgroups of Q, each of which has a system that is partition regular over it but not over its predecessors. We actually prove something stronger: our main result is that if R and S are subrings of Q with R not contained in S, then there is a system that is partition regular over R but not over S. This implies, for example, that the chain above may be taken to be uncountable. (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

Free