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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

   

 

Ideal games and Ramsey sets


Authors: Carlos Di Prisco, José G. Mijares and Carlos Uzcátegui
Journal: Proc. Amer. Math. Soc. 140 (2012), 2255-2265
MSC (2010): Primary 05D10; Secondary 03E02
Published electronically: November 1, 2011
MathSciNet review: 2898689
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Abstract: It is shown that Matet's characterization of the Ramsey property relative to a selective co-ideal $ \mathcal {H}$, in terms of games of Kastanas, still holds if we consider semiselectivity instead of selectivity. Moreover, we prove that a co-ideal $ \mathcal {H}$ is semiselective if and only if Matet's game-theoretic characterization of the $ \mathcal {H}$-Ramsey property holds. This lifts Kastanas's characterization of the classical Ramsey property to its optimal setting, from the point of view of the local Ramsey theory, and gives a game-theoretic counterpart to a theorem of Farah, asserting that a co-ideal $ \mathcal {H}$ is semiselective if and only if the family of $ \mathcal {H}$-Ramsey subsets of $ \mathbb{N}^{[\infty ]}$ coincides with the family of those sets having the abstract $ \mathcal {H}$-Baire property. Finally, we show that under suitable assumptions, for every semiselective co-ideal $ \mathcal H$ all sets of real numbers are $ \mathcal H$-Ramsey.


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Additional Information

Carlos Di Prisco
Affiliation: Instituto Venezolano de Investigaciones Científicas y Escuela de Matemática, Universidad Central de Venezuela, Caracas, Venezuela
Email: cdiprisc@ivic.gob.ve

José G. Mijares
Affiliation: Instituto Venezolano de Investigaciones Científicas y Escuela de Matemática, Universidad Central de Venezuela, Caracas, Venezuela
Address at time of publication: Departamento de Matematicas, Pontificia Universidad Javeriana, Bogota, Colombia
Email: jmijares@ivic.gob.ve, jose.mijares@ciens.ucv.ve, jmijares@javeriana.edu.co

Carlos Uzcátegui
Affiliation: Departamento de Matemáticas, Facultad de Ciencias, Universidad de Los Andes, Mérida, Venezuela
Email: uzca@ula.ve

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-11090-6
Keywords: Semiselective co-ideal, Ramsey theory, Kastanas games, Banach-Mazur games
Received by editor(s): September 19, 2010
Received by editor(s) in revised form: February 19, 2011
Published electronically: November 1, 2011
Communicated by: Julia Knight
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.



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