Several types of polarized partition relations are considered. In particular we deal with partitions defined on cartesian products of more than two factors. MSC: 03E05.

Given a topological Ramsey space math formula, we extend the notion of semiselective coideal to sets math formula and study conditions for math formula that will enable us to make the structure math formula a Ramsey space and also study forcing notions related to math formula which will satisfy abstract versions of interesting properties of the corresponding forcing notions in the realm of Ellentuck's space. This extends results from to the most general context of topological Ramsey spaces. As applications, we (...) prove that for every topological Ramsey space math formula, under suitable large cardinal hypotheses every semiselective ultrafilter math formula is generic over math formula; and that given a semiselective coideal math formula, every definable subset of math formula is math formula-Ramsey. This generalizes the corresponding results for the case when math formula is equal to Ellentuck's space. (shrink)

We consider several kinds of partition relations on the set ${\mathbb{R}}$ of real numbers and its powers, as well as their parameterizations with the set ${[\mathbb{N}]^{\mathbb{N}}}$ of all infinite sets of natural numbers, and show that they hold in some models of set theory. The proofs use generic absoluteness, that is, absoluteness under the required forcing extensions. We show that Solovay models are absolute under those forcing extensions, which yields, for instance, that in these models for every well ordered partition (...) of ${\mathbb{R}^\mathbb{N}}$ there is a sequence of perfect sets whose product lies in one piece of the partition. Moreover, for every finite partition of ${[\mathbb{N}]^{\mathbb{N}} \times \mathbb{R}^{\mathbb{N}}}$ there is ${X \in [\mathbb{N}]^{\mathbb{N}}}$ and a sequence ${\{P_{k} : k \in \mathbb{N}\}}$ of perfect sets such that the product ${[X]^{\mathbb{N}} \times \prod_{k}^{\infty}P_{k}}$ lies in one piece of the partition, where ${[X]^{\mathbb{N}}}$ is the set of all infinite subsets of X. The proofs yield the same results for Borel partitions in ZFC, and for more complex partitions in any model satisfying a certain degree of generic absoluteness. (shrink)

The Continuum Hypothesis has motivated a considerable part of the development of axiomatic set theory for over a century. We present, in a very schematic way, some of the results that give information related to Cantor’s Continuum Problem.

We prove that the least σ-algebra containing the projective sets and closed under projections is exactly the collection of hyperprojective sets which, with their complements, can be inductively defined with real parameters by an induction of countable length. This provides a construction principle for this natural class of hyperprojective sets.

We study several perfect set properties of the Baire space which follow from the Ramsey property ω→(ω) ω . In particular we present some independence results which complete the picture of how these perfect set properties relate to each other.

El objetivo de este trabajo es presentar en un solo cuerpo tres maneras de relativizar (o generalizar) el concepto de conjunto constructible de Gödel que no suelen aparecer juntas en la literatura especializada y que son importantes en la Teoría de Conjuntos, por ejemplo para resolver problemas de consistencia o independencia. Presentamos algunos modelos resultantes de las diferentes formas de relativizar el concepto de constructibilidad, sus propiedades básicas y algunas formas débiles del Axioma de Elección válidas o no válidas en (...) ellas. (shrink)

The Principle of Ariadne, formulated in 1988 ago by Walter Carnielli and Carlos Di Prisco and later published in 1993, is an infinitary principle that is independent of the Axiom of Choice in ZF, although it can be consistently added to the remaining ZF axioms. The present paper surveys, and motivates, the foundational importance of the Principle of Ariadne and proposes the Ariadne Game, showing that the Principle of Ariadne, corresponds precisely to a winning strategy for the Ariadne (...) Game. Some relations to other alternative. set-theoretical principles are also briefly discussed. (shrink)

Effective vaccines for COVID‐19 are already available to humankind. In Peru, 86 million doses were administered to cover the demand for 33 million Peruvian people. Hence, vaccination has been prioritized in groups: health personnel, subjects with pre‐existing health conditions and those over 65 years of age. However, given the social problems and the public health situation in Peru, this work defends that the priority of vaccination should be focused on the population living in extreme poverty. The method used was an (...) ethical argumentation on the distribution of scarce antiSARS‐CoV2 vaccine in Peru. This argument is based on the analysis of the Peruvian population living in extreme poverty, which presents different layers of vulnerability, and that, in the face of an eventual SARS‐CoV2 infection, these would be exacerbated one after the other, through a cascade effect. This scenario would give rise to new vulnerabilities to those already existing, causing greater damage. Vaccination efforts on this key population would give them the opportunity to continue to find ways to bring food to their homes, significantly reducing the risk of contagion in their environment and mitigating the devastating effect of the local diseases to which they are already exposed. Four objections related to this argument are raised with their corresponding responses. Priority access to the vaccine would significantly reduce the humanitarian harm to people living in extreme poverty, prevailing the principles of justice and equity. (shrink)

La humanidad ya dispone de vacunas eficaces contra el COVID‐19. En Perú se administraron 86 millones de dosis para cubrir la demanda de 33 millones de peruanos. Para ello, se ha priorizado la vacunación en grupos clave: personal de salud, sujetos con condiciones de salud preexistentes y mayores de 65 años. Sin embargo, dada la problemática social y la situación de la salud pública en Perú, este trabajo defiende que la prioridad de la vacunación debe centrarse en la población que (...) vive en extrema pobreza. El método utilizado fue una argumentación ética sobre la distribución de la vacuna contra el COVID‐19 en Perú. Esta argumentación se basa en el análisis de la población peruana que vive en extrema pobreza, la cual presenta diferentes estratos de vulnerabilidad, y que, ante una eventual infección por SARS‐CoV‐2, se irían agravando uno tras otro, a través de un efecto en cascada. Este escenario daría lugar a nuevas vulnerabilidades de las ya existentes, causando mayores daños. Los esfuerzos de vacunación en esta población clave les brindaría oportunidad de seguir encontrando formas de llevar alimentos a sus hogares, reduciendo significativamente el riesgo de contagio en su entorno y mitigando el efecto devastador de las enfermedades locales a las que ya está expuesta. Se plantean cuatro objeciones relacionadas con este argumento, con sus correspondientes respuestas. El acceso prioritario a la vacuna reduciría significativamente el daño a las personas que viven en la extrema pobreza, haciendo prevalecer los principios de justicia y equidad. (shrink)