Rosenthal families, pavings, and generic cardinal invariants

Koszmider, Piotr; Martínez-Celis, Arturo

doi:10.1090/proc/15252

Rosenthal families, pavings, and generic cardinal invariants
HTML articles powered by AMS MathViewer

by Piotr Koszmider and Arturo Martínez-Celis PDF

Proc. Amer. Math. Soc. 149 (2021), 1289-1303 Request permission

Abstract:

Following D. Sobota we call a family $\mathcal F$ of infinite subsets of $\mathbb {N}$ a Rosenthal family if it can replace the family of all infinite subsets of $\mathbb {N}$ in the classical Rosenthal lemma concerning sequences of measures on pairwise disjoint sets. We resolve two problems on Rosenthal families: every ultrafilter is a Rosenthal family and the minimal size of a Rosenthal family is exactly equal to the reaping cardinal $\mathfrak r$. This is achieved through analyzing nowhere reaping families of subsets of $\mathbb {N}$ and through applying a paving lemma which is a consequence of a paving lemma concerning linear operators on $\ell _1^n$ due to Bourgain. We use connections of the above results with free set results for functions on $\mathbb {N}$ and with linear operators on $c_0$ to determine the values of several other derived cardinal invariants.

References

Jason Aubrey, Combinatorics for the dominating and unsplitting numbers, J. Symbolic Logic 69 (2004), no. 2, 482–498. MR 2058185, DOI 10.2178/jsl/1082418539
Bohuslav Balcar and Petr Simon, On minimal $\pi$-character of points in extremally disconnected compact spaces, Topology Appl. 41 (1991), no. 1-2, 133–145. MR 1129703, DOI 10.1016/0166-8641(91)90105-U
Andreas Blass, Combinatorial cardinal characteristics of the continuum, Handbook of set theory. Vols. 1, 2, 3, Springer, Dordrecht, 2010, pp. 395–489. MR 2768685, DOI 10.1007/978-1-4020-5764-9_{7}
David Booth, Ultrafilters on a countable set, Ann. Math. Logic 2 (1970/71), no. 1, 1–24. MR 277371, DOI 10.1016/0003-4843(70)90005-7
Jean Bourgain, New classes of ${\cal L}^{p}$-spaces, Lecture Notes in Mathematics, vol. 889, Springer-Verlag, Berlin-New York, 1981. MR 639014
J. Bourgain and L. Tzafriri, Restricted invertibility of matrices and applications, Analysis at Urbana, Vol. II (Urbana, IL, 1986–1987) London Math. Soc. Lecture Note Ser., vol. 138, Cambridge Univ. Press, Cambridge, 1989, pp. 61–107. MR 1009186
Marcin Bownik, The Kadison-Singer problem, Frames and harmonic analysis, Contemp. Math., vol. 706, Amer. Math. Soc., [Providence], RI, [2018] ©2018, pp. 63–92. MR 3796632, DOI 10.1090/conm/706/14218
N. G. de Bruijn and P. Erdös, A colour problem for infinite graphs and a problem in the theory of relations, Nederl. Akad. Wetensch. Proc. Ser. A. 54 = Indagationes Math. 13 (1951), 369–373. MR 0046630
David Chodounský and Osvaldo Guzmán, There are no P-points in Silver extensions, Israel J. Math. 232 (2019), no. 2, 759–773. MR 3990958, DOI 10.1007/s11856-019-1886-2
J. Diestel and J. J. Uhl Jr., Vector measures, Mathematical Surveys, No. 15, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis. MR 0453964
Joseph Diestel, Sequences and series in Banach spaces, Graduate Texts in Mathematics, vol. 92, Springer-Verlag, New York, 1984. MR 737004, DOI 10.1007/978-1-4612-5200-9
P. Erdös, Some remarks on set theory, Proc. Amer. Math. Soc. 1 (1950), 127–141. MR 35809, DOI 10.1090/S0002-9939-1950-0035809-8
Ilijas Farah, Semiselective coideals, Mathematika 45 (1998), no. 1, 79–103. MR 1644345, DOI 10.1112/S0025579300014054
M. Goldstern and S. Shelah, Ramsey ultrafilters and the reaping number—$\textrm {Con}({\mathfrak {r}}<{\mathfrak {u}})$, Ann. Pure Appl. Logic 49 (1990), no. 2, 121–142. MR 1077075, DOI 10.1016/0168-0072(90)90063-8
A. Grothendieck, Sur les applications linéaires faiblement compactes d’espaces du type $C(K)$, Canad. J. Math. 5 (1953), 129–173 (French). MR 58866, DOI 10.4153/cjm-1953-017-4
Michael Hrušák, David Meza-Alcántara, and Hiroaki Minami, Pair-splitting, pair-reaping and cardinal invariants of $F_\sigma$-ideals, J. Symbolic Logic 75 (2010), no. 2, 661–677. MR 2648159, DOI 10.2178/jsl/1268917498
William B. Johnson and Gideon Schechtman, Finite dimensional subspaces of $L_p$, Handbook of the geometry of Banach spaces, Vol. I, North-Holland, Amsterdam, 2001, pp. 837–870. MR 1863707, DOI 10.1016/S1874-5849(01)80021-8
Péter Komjáth and Vilmos Totik, Problems and theorems in classical set theory, Problem Books in Mathematics, Springer, New York, 2006. MR 2220838
Joram Lindenstrauss, On complemented subspaces of $m$, Israel J. Math. 5 (1967), 153–156. MR 222616, DOI 10.1007/BF02771101
Adam W. Marcus, Daniel A. Spielman, and Nikhil Srivastava, Interlacing families II: Mixed characteristic polynomials and the Kadison-Singer problem, Ann. of Math. (2) 182 (2015), no. 1, 327–350. MR 3374963, DOI 10.4007/annals.2015.182.1.8
A. Pełczyński, Projections in certain Banach spaces, Studia Math. 19 (1960), 209–228. MR 126145, DOI 10.4064/sm-19-2-209-228
R. S. Phillips, On linear transformations, Trans. Amer. Math. Soc. 48 (1940), 516–541. MR 4094, DOI 10.1090/S0002-9947-1940-0004094-3
Haskell P. Rosenthal, On complemented and quasi-complemented subspaces of quotients of $C(S)$ for Stonian $S$, Proc. Nat. Acad. Sci. U.S.A. 60 (1968), 1165–1169. MR 231185, DOI 10.1073/pnas.60.4.1165
Haskell P. Rosenthal, On relatively disjoint families of measures, with some applications to Banach space theory, Studia Math. 37 (1970), 13–36. MR 270122, DOI 10.4064/sm-37-1-13-36
Damian Sobota, The Nikodym property and cardinal characteristics of the continuum, Ann. Pure Appl. Logic 170 (2019), no. 1, 1–35. MR 3870039, DOI 10.1016/j.apal.2018.07.003
Damian Sobota, Families of sets related to Rosenthal’s lemma, Arch. Math. Logic 58 (2019), no. 1-2, 53–69. MR 3902804, DOI 10.1007/s00153-018-0621-8
Angus Ellis Taylor and David C. Lay, Introduction to functional analysis, 2nd ed., John Wiley & Sons, New York-Chichester-Brisbane, 1980. MR 564653
Edward L. Wimmers, The Shelah $P$-point independence theorem, Israel J. Math. 43 (1982), no. 1, 28–48. MR 728877, DOI 10.1007/BF02761683