Distinguishing subgroups of the rationals by their Ramsey properties

Ben Barber; Neil Hindman; Imre Leader; Dona Strauss

doi:10.1016/j.jcta.2014.10.002

Distinguishing subgroups of the rationals by their Ramsey properties

Ben Barber, Neil Hindman, Imre Leader, Dona Strauss

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Abstract

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.

Original language	English
Pages (from-to)	93–104
Number of pages	12
Journal	Journal of Combinatorial Theory, Series A
Volume	129
Early online date	15 Oct 2014
DOIs	https://doi.org/10.1016/j.jcta.2014.10.002
Publication status	Published - 1 Jan 2015

Keywords

Partion regular
Rationals
Subgroups
Central sets

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10.1016/j.jcta.2014.10.002

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Accepted author manuscript, 193 KBLicence: CC BY-NC-ND

http://arxiv.org/abs/1405.0460

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title = "Distinguishing subgroups of the rationals by their Ramsey properties",

abstract = "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.",

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AU - Barber, Ben

AU - Hindman, Neil

AU - Leader, Imre

AU - Strauss, Dona

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Y1 - 2015/1/1

N2 - 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.

AB - 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.

KW - Partion regular

KW - Rationals

KW - Subgroups

KW - Central sets

U2 - 10.1016/j.jcta.2014.10.002

DO - 10.1016/j.jcta.2014.10.002

M3 - Article (Academic Journal)

SN - 0097-3165

VL - 129

SP - 93

EP - 104

JO - Journal of Combinatorial Theory, Series A

JF - Journal of Combinatorial Theory, Series A

ER -

Distinguishing subgroups of the rationals by their Ramsey properties

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Keywords

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