The space of infinite partitions of $\mathbb N$ as a topological Ramsey space
DOI:
https://doi.org/10.33044/revuma.2869Abstract
The Ramsey theory of the space of equivalence relations with infinite quotients defined on the set $\mathbb{N}$ of natural numbers is an interesting field of research. We view this space as a topological Ramsey space $(\mathcal{E}_\infty,\leq,r)$ and present a game theoretic characterization of the Ramsey property of subsets of $\mathcal{E}_{\infty}$. We define a notion of coideal and consider the Ramsey property of subsets of $\mathcal{E}_\infty$ localized on a coideal $\mathcal{H}\subseteq \mathcal{E}_{\infty}$. Conditions a coideal $\mathcal{H}$ should satisfy to make the structure $(\mathcal{E}_{\infty},\mathcal{H},\leq, r)$ a Ramsey space are presented. Forcing notions related to a coideal $\mathcal{H}$ and their main properties are analyzed.
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References
T. J. Carlson and S. G. Simpson, A dual form of Ramsey's theorem, Adv. in Math. 53 (1984), no. 3, 265–290. MR 0753869.
T. J. Carlson and S. G. Simpson, Topological Ramsey theory, in Mathematics of Ramsey Theory, 172–183, Algorithms Combin., 5, Springer, Berlin, 1990. MR 1083600.
J. Cichoń, A. Krawczyk, B. Majcher-Iwanow and B. Wȩglorz, Dualization of the van Douwen diagram, J. Symbolic Logic 65 (2000), no. 2, 959–968. MR 1771096.
C. Di Prisco, J. G. Mijares and J. Nieto, Local Ramsey theory: an abstract approach, MLQ Math. Log. Q. 63 (2017), no. 5, 384–396. MR 3748482.
C. Di Prisco, J. G. Mijares and C. Uzcátegui, Ideal games and Ramsey sets, Proc. Amer. Math. Soc. 140 (2012), no. 7, 2255–2265. MR 2898689.
E. Ellentuck, A new proof that analytic sets are Ramsey, J. Symbolic Logic 39 (1974), 163–165. MR 0349393.
I. Farah, Semiselective coideals, Mathematika 45 (1998), no. 1, 79–103. MR 1644345.
F. Galvin, A generalization of Ramsey's theorem, Notices Amer. Math. Soc. 15 (1968), 548. https://www.ams.org/journals/notices/196804/196804FullIssue.pdf.
F. Galvin and K. Prikry, Borel sets and Ramsey's theorem, J. Symbolic Logic 38 (1973), 193–198. MR 0337630.
L. Halbeisen, Symmetries between two Ramsey properties, Arch. Math. Logic 37 (1998), no. 4, 241–260. MR 1635558.
L. Halbeisen, On shattering, splitting and reaping partitions, Math. Logic Quart. 44 (1998), no. 1, 123–134. MR 1601919.
L. Halbeisen, Ramseyan ultrafilters, Fund. Math. 169 (2001), no. 3, 233–248. MR 1852127.
L. J. Halbeisen, Combinatorial Set Theory: With a Gentle Introduction to Forcing, Second Edition, Springer Monographs in Mathematics, Springer, Cham, 2017. MR 3751612.
L. Halbeisen and P. Matet, A generalization of the dual Ellentuck theorem, Arch. Math. Logic 42 (2003), no. 2, 103–128. MR 1961873.
T. Jech, Set Theory, the third millennium edition, revised and expanded, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. MR 1940513.
I. G. Kastanas, On the Ramsey property for sets of reals, J. Symbolic Logic 48 (1983), no. 4, 1035–1045 (1984). MR 0727792.
P. Matet, Partitions and filters, J. Symbolic Logic 51 (1986), no. 1, 12–21. MR 0830067.
P. Matet, Some filters of partitions, J. Symbolic Logic 53 (1988), no. 2, 540–553. MR 0947858.
P. Matet, Happy families and completely Ramsey sets, Arch. Math. Logic 32 (1993), no. 3, 151–171. MR 1201647.
A. R. D. Mathias, Happy families, Ann. Math. Logic 12 (1977), no. 1, 59–111. MR 0491197.
J. G. Mijares, A notion of selective ultrafilter corresponding to topological Ramsey spaces, MLQ Math. Log. Q. 53 (2007), no. 3, 255–267. MR 2330595.
F. P. Ramsey, On a problem of formal logic, Proc. London Math. Soc. (2) 30 (1929), no. 4, 264–286. MR 1576401.
S. Todorcevic, Introduction to Ramsey Spaces, Annals of Mathematics Studies, 174, Princeton University Press, Princeton, NJ, 2010. MR 2603812.
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