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/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

We work within the framework of the Alpha-Theory introduced by Benci and DiNasso. The Alpha-Theory postulates a few natural properties for an infinite"ideal" number $\alpha$. The formulation provides an elementary axiomatics forthe methods of abstract ultra-Ramsey theory.  The main results are Theorem 10, Theorem 57, Theorem 67 and Theorem 73.Theorem 10 is an infinite-dimensional extension of the celebrated Ramsey'sTheorem. We show that corollaries of this result include the Galvin-PirkyTheorem, the Silver Theorem and the $\vec{\alpha}$-Ellentuck Theorem. We provethat, under the assumption of the $\mathfrak{c}^{+}$-enlarging property, the$\vec{\alpha}$-Ellentuck Theorem is equivalent to the Ultra-Ellentuck Theoremof Todorcevic.  Theorem 57 is an abstraction of Theorem 10 to the setting of triples$(\mathcal{R},\le,r)$ where $\mathcal{R}\not=\emptyset$, $\le$ is a quasi-orderon $\mathcal{R}$ and $r$ is a function with domain $\mathbb{N}\times\mathcal{R}$. We use Theorem 57 to develop the Abstract$\vec{\alpha}$-Ellentuck Theorem, Theorem 67, and the Abstract Ultra-EllentuckTheorem, Theorem 73, extending the Abstract Ellentuck Theorem along the samelines as the $\vec{\alpha}$-Ellentuck Theorem and Ultra-Ellentuck Theoremextend the Ellentuck Theorem, respectively. We conclude with some examplesillustrating the theory and an open question related to the local Ramsey theorydeveloped by Di Prisco, Mijares and Nieto.

From abstract alpha-Ramsey theory to abstract ultra-Ramsey theory