Infinite dimensional perfect set theorems

Mátrai, Tamás

doi:10.1090/S0002-9947-2012-05468-7

Infinite dimensional perfect set theorems
HTML articles powered by AMS MathViewer

by Tamás Mátrai PDF

Trans. Amer. Math. Soc. 365 (2013), 23-58 Request permission

Abstract:

What largeness and structural assumptions on $A \subseteq [\mathbb {R}]^{\omega }$ can guarantee the existence of a non-empty perfect set $P \subseteq \mathbb {R}$ such that $[P]^{\omega } \subseteq A$? Such a set $P$ is called $A$-homogeneous. We show that even if $A$ is open, in general it is independent of ZFC whether for a cardinal $\kappa$, the existence of an $A$-homogeneous set $H \in [\mathbb {R}]^{\kappa }$ implies the existence of a non-empty perfect $A$-homogeneous set.

On the other hand, we prove an infinite dimensional analogue of Mycielski’s Theorem: if $A$ is large in the sense of a suitable Baire category-like notion, then there exists a non-empty perfect $A$-homogeneous set. We introduce fusion games to prove this and other infinite dimensional perfect set theorems.

Finally we apply this theory to show that it is independent of ZFC whether Tukey reductions of the maximal analytic cofinal type can be witnessed by definable Tukey maps.

References

Tomek Bartoszynski, Invariants of measure and category, Handbook of set theory. Vols. 1, 2, 3, Springer, Dordrecht, 2010, pp. 491–555. MR 2768686, DOI 10.1007/978-1-4020-5764-9_{8}
M. R. Burke, A theorem of Friedman on rectangle inclusion and its consequences, note.
Douglas Cenzer and R. Daniel Mauldin, On the Borel class of the derived set operator, Bull. Soc. Math. France 110 (1982), no. 4, 357–380 (English, with French summary). MR 694756, DOI 10.24033/bsmf.1968
Douglas Cenzer and R. Daniel Mauldin, On the Borel class of the derived set operator. II, Bull. Soc. Math. France 111 (1983), no. 4, 367–372 (English, with French summary). MR 763550, DOI 10.24033/bsmf.1994
D. H. Fremlin, The partially ordered sets of measure theory and Tukey’s ordering, Note Mat. 11 (1991), 177–214. Dedicated to the memory of Professor Gottfried Köthe. MR 1258546
D. H. Fremlin, Families of compact sets and Tukey’s ordering, Atti Sem. Mat. Fis. Univ. Modena 39 (1991), no. 1, 29–50. MR 1111757
H. Friedman, Rectangle inclusion problems, note.
Leo Harrington and Saharon Shelah, Counting equivalence classes for co-$\kappa$-Souslin equivalence relations, Logic Colloquium ’80 (Prague, 1980) Studies in Logic and the Foundations of Mathematics, vol. 108, North-Holland, Amsterdam-New York, 1982, pp. 147–152. MR 673790
Thomas Jech, Set theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. The third millennium edition, revised and expanded. MR 1940513
Vassilis Kanellopoulos, Ramsey families of subtrees of the dyadic tree, Trans. Amer. Math. Soc. 357 (2005), no. 10, 3865–3886. MR 2159691, DOI 10.1090/S0002-9947-05-03968-1
Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597, DOI 10.1007/978-1-4612-4190-4
Wiesław Kubiś, Perfect cliques and $G_\delta$ colorings of Polish spaces, Proc. Amer. Math. Soc. 131 (2003), no. 2, 619–623. MR 1933354, DOI 10.1090/S0002-9939-02-06584-X
Wiesław Kubiś and Saharon Shelah, Analytic colorings, Ann. Pure Appl. Logic 121 (2003), no. 2-3, 145–161. MR 1982945, DOI 10.1016/S0168-0072(02)00110-0
Kenneth Kunen, Set theory, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland Publishing Co., Amsterdam-New York, 1980. An introduction to independence proofs. MR 597342
A. Louveau, S. Shelah, and B. Veličković, Borel partitions of infinite subtrees of a perfect tree, Ann. Pure Appl. Logic 63 (1993), no. 3, 271–281. MR 1237234, DOI 10.1016/0168-0072(93)90151-3
Alain Louveau and Boban Velickovic, Analytic ideals and cofinal types, Ann. Pure Appl. Logic 99 (1999), no. 1-3, 171–195. MR 1708151, DOI 10.1016/S0168-0072(98)00065-7
Krzysztof Mazur, A modification of Louveau and Veličkovič’s construction for $F_\sigma$-ideals, Proc. Amer. Math. Soc. 128 (2000), no. 5, 1475–1479. MR 1626442, DOI 10.1090/S0002-9939-00-05061-9
Arnold W. Miller, Mapping a set of reals onto the reals, J. Symbolic Logic 48 (1983), no. 3, 575–584. MR 716618, DOI 10.2307/2273449
Arnold W. Miller, On the length of Borel hierarchies, Ann. Math. Logic 16 (1979), no. 3, 233–267. MR 548475, DOI 10.1016/0003-4843(79)90003-2
Jan Mycielski, Independent sets in topological algebras, Fund. Math. 55 (1964), 139–147. MR 173645, DOI 10.4064/fm-55-2-139-147
M. Sabok, Forcing, games and families of closed sets, Trans. Amer. Math. Soc., to appear.
M. Sabok, On Idealized Forcing, Instytut Matematyczny Uniwersytetu Wrocławskiego, Ph.D. thesis.
Saharon Shelah, Borel sets with large squares, Fund. Math. 159 (1999), no. 1, 1–50. MR 1669643, DOI 10.4064/fm-159-1-1-50
Sławomir Solecki, Covering analytic sets by families of closed sets, J. Symbolic Logic 59 (1994), no. 3, 1022–1031. MR 1295987, DOI 10.2307/2275926
Sławomir Solecki and Stevo Todorcevic, Cofinal types of topological directed orders, Ann. Inst. Fourier (Grenoble) 54 (2004), no. 6, 1877–1911 (2005) (English, with English and French summaries). MR 2134228, DOI 10.5802/aif.2070
Stevo Todorcevic, A classification of transitive relations on $\omega _1$, Proc. London Math. Soc. (3) 73 (1996), no. 3, 501–533. MR 1407459, DOI 10.1112/plms/s3-73.3.501
Stevo Todorčević, Directed sets and cofinal types, Trans. Amer. Math. Soc. 290 (1985), no. 2, 711–723. MR 792822, DOI 10.1090/S0002-9947-1985-0792822-9
Stevo Todorcevic, Introduction to Ramsey spaces, Annals of Mathematics Studies, vol. 174, Princeton University Press, Princeton, NJ, 2010. MR 2603812, DOI 10.1515/9781400835409
John W. Tukey, Convergence and Uniformity in Topology, Annals of Mathematics Studies, No. 2, Princeton University Press, Princeton, N. J., 1940. MR 0002515
Jindřich Zapletal, Forcing idealized, Cambridge Tracts in Mathematics, vol. 174, Cambridge University Press, Cambridge, 2008. MR 2391923, DOI 10.1017/CBO9780511542732
J. Zapletal, Homogeneous sets of positive outer measure, preprint.