English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Open

Presents the pdf analysis as premium feature

PDF Analysis

Insights

Presents the summarisation as premium feature

Summarisation

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

A system of linear equations with integer coefficients is partition regularover a subset S of the reals if, whenever S\{0} is finitely coloured, there isa solution to the system contained in one colour class. It has been known forsome time that there is an infinite system of linear equations that ispartition regular over R but not over Q, and it was recently shown (answering along-standing open question) that one can also distinguish Q from Z in thisway.  Our aim is to show that the transition from Z to Q is not sharp: there is aninfinite chain of subgroups of Q, each of which has a system that is partitionregular over it but not over its predecessors. We actually prove somethingstronger: our main result is that if R and S are subrings of Q with R notcontained in S, then there is a system that is partition regular over R but notover S. This implies, for example, that the chain above may be taken to beuncountable.

Distinguishing subgroups of the rationals by their Ramsey properties