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Distinguishing subgroups of the rationals by their Ramsey properties.
- Source :
-
Journal of Combinatorial Theory - Series A . Jan2015, Vol. 129, p93-104. 12p. - Publication Year :
- 2015
-
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. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 00973165
- Volume :
- 129
- Database :
- Academic Search Index
- Journal :
- Journal of Combinatorial Theory - Series A
- Publication Type :
- Academic Journal
- Accession number :
- 99405659
- Full Text :
- https://doi.org/10.1016/j.jcta.2014.10.002