Selective ultrafilters on FIN

Zheng, Yuan Yuan

doi:10.1090/proc/13644

Selective ultrafilters on FIN
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Proc. Amer. Math. Soc. 145 (2017), 5071-5086 Request permission

Abstract:

We consider selective ultrafilters on the collection $\mathrm {FIN}$ of all finite nonempty subsets of $\mathbb {N}$. If countable-support side-by-side Sacks forcing is applied, then every selective ultrafilter in the ground model generates a selective ultrafilter in the extension. We also show that selective ultrafilters localize the Parametrized Milliken Theorem, and that selective ultrafilters are Ramsey.

References

James E. Baumgartner, Sacks forcing and the total failure of Martin’s axiom, Topology Appl. 19 (1985), no. 3, 211–225. MR 794485, DOI 10.1016/0166-8641(85)90002-1
James E. Baumgartner and Richard Laver, Iterated perfect-set forcing, Ann. Math. Logic 17 (1979), no. 3, 271–288. MR 556894, DOI 10.1016/0003-4843(79)90010-X
A. Blass, Ultrafilters related to Hindman’s finite-unions theorem and its extensions, In Logic and combinatorics (Arcata, Calif., 1985), volume 65 of Contemp. Math., pages 89–124. Amer. Math. Soc., Providence, RI, 1987.
Andreas Blass, Selective ultrafilters and homogeneity, Ann. Pure Appl. Logic 38 (1988), no. 3, 215–255. MR 942525, DOI 10.1016/0168-0072(88)90027-9
C. A. Di Prisco, J. G. Mijares, and J. E. Nieto, Local Ramsey Theory. An Abstract Approach. Preprint.
N. Dobrinen, J. G. Mijares, and T. Trujillo, Topological Ramsey spaces from Fraisse classes, Ramsey-classification theorems, and initial structures in the Tukey types of p-points, Archive for Mathematical Logic, special issue in honor of James E. Baumgartner, (2017), 1-50, DOI 10.1007/s00153-017-0540-0.
Natasha Dobrinen and Stevo Todorcevic, A new class of Ramsey-classification theorems and their applications in the Tukey theory of ultrafilters, Part 2, Trans. Amer. Math. Soc. 367 (2015), no. 7, 4627–4659. MR 3335396, DOI 10.1090/S0002-9947-2014-06122-9
Erik Ellentuck, A new proof that analytic sets are Ramsey, J. Symbolic Logic 39 (1974), 163–165. MR 349393, DOI 10.2307/2272356
Ilijas Farah, Semiselective coideals, Mathematika 45 (1998), no. 1, 79–103. MR 1644345, DOI 10.1112/S0025579300014054
D. J. Fernández-Bretón and M. Hrušák, A weak diamond principle and union ultrafilters, Unpublished notes.
Fred Galvin and Karel Prikry, Borel sets and Ramsey’s theorem, J. Symbolic Logic 38 (1973), 193–198. MR 337630, DOI 10.2307/2272055
Richard Laver, Products of infinitely many perfect trees, J. London Math. Soc. (2) 29 (1984), no. 3, 385–396. MR 754925, DOI 10.1112/jlms/s2-29.3.385
A. R. D. Mathias, On a generalisation of Ramsey’s theorem, Notices of the American Mathematical Society, 15:931, 1968.
José G. Mijares, A notion of selective ultrafilter corresponding to topological Ramsey spaces, MLQ Math. Log. Q. 53 (2007), no. 3, 255–267. MR 2330595, DOI 10.1002/malq.200510045
José G. Mijares, Parametrizing the abstract Ellentuck theorem, Discrete Math. 307 (2007), no. 2, 216–225. MR 2285192, DOI 10.1016/j.disc.2006.06.022
Keith R. Milliken, Ramsey’s theorem with sums or unions, J. Combinatorial Theory Ser. A 18 (1975), 276–290. MR 373906, DOI 10.1016/0097-3165(75)90039-4
Yiannis N. Moschovakis, Descriptive set theory, 2nd ed., Mathematical Surveys and Monographs, vol. 155, American Mathematical Society, Providence, RI, 2009. MR 2526093, DOI 10.1090/surv/155
C. St. J. A. Nash-Williams, On well-quasi-ordering transfinite sequences, Proc. Cambridge Philos. Soc. 61 (1965), 33–39. MR 173640, DOI 10.1017/s0305004100038603
Pavel Pudlák and Vojtěch Rödl, Partition theorems for systems of finite subsets of integers, Discrete Math. 39 (1982), no. 1, 67–73. MR 677888, DOI 10.1016/0012-365X(82)90041-3
Gerald E. Sacks, Forcing with perfect closed sets, Axiomatic Set Theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967) Amer. Math. Soc., Providence, R.I., 1971, pp. 331–355. MR 0276079
Jack Silver, Every analytic set is Ramsey, J. Symbolic Logic 35 (1970), 60–64. MR 332480, DOI 10.2307/2271156
Stevo Todorcevic, Introduction to Ramsey spaces, Annals of Mathematics Studies, vol. 174, Princeton University Press, Princeton, NJ, 2010. MR 2603812, DOI 10.1515/9781400835409
Timothy Onofre Trujillo, Topological Ramsey spaces, associated ultrafilters, and their applications to the Tukey Theory of ultrafilters and Dedekind cuts of nonstandard arithmetic, ProQuest LLC, Ann Arbor, MI, 2014. Thesis (Ph.D.)–University of Denver. MR 3301648
Timothy Trujillo, Selective but not Ramsey, Topology Appl. 202 (2016), 61–69. MR 3464149, DOI 10.1016/j.topol.2015.12.088