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

Abstract

We work within the framework of the Alpha-Theory introduced by Benci and Di Nasso. The Alpha-Theory postulates a few natural properties for an infinite "ideal" number $α$. The formulation provides an elementary axiomatics for the 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's Theorem. We show that corollaries of this result include the Galvin-Pirky Theorem, the Silver Theorem and the $\vecα$-Ellentuck Theorem. We prove that, under the assumption of the $\mathfrak{c}^{+}$-enlarging property, the $\vecα$-Ellentuck Theorem is equivalent to the Ultra-Ellentuck Theorem of 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-order on $\mathcal{R}$ and $r$ is a function with domain $\mathbb{N}\times \mathcal{R}$. We use Theorem 57 to develop the Abstract $\vecα$-Ellentuck Theorem, Theorem 67, and the Abstract Ultra-Ellentuck Theorem, Theorem 73, extending the Abstract Ellentuck Theorem along the same lines as the $\vecα$-Ellentuck Theorem and Ultra-Ellentuck Theorem extend the Ellentuck Theorem, respectively. We conclude with some examples illustrating the theory and an open question related to the local Ramsey theory developed by Di Prisco, Mijares and Nieto.