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On two problems of Erdos and Hechler: New methods in singular madness


Authors: Menachem Kojman, Wieslaw Kubis and Saharon Shelah
Journal: Proc. Amer. Math. Soc. 132 (2004), 3357-3365
MSC (2000): Primary 03E10, 03E04, 03E17, 03E35; Secondary 03E55, 03E50
DOI: https://doi.org/10.1090/S0002-9939-04-07580-X
Published electronically: June 21, 2004
MathSciNet review: 2073313
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Abstract: For an infinite cardinal $\mu$, $\operatorname{MAD}(\mu)$denotes the set of all cardinalities of nontrivial maximal almost disjoint families over $\mu$.

Erdos and Hechler proved in 1973 the consistency of $\mu\in \operatorname{MAD}(\mu)$for a singular cardinal $\mu$ and asked if it was ever possible for a singular $\mu$ that $\mu\notin \operatorname{MAD}(\mu)$, and also whether $2^{\operatorname{cf}\mu} <\mu \Longrightarrow \mu\in \operatorname{MAD}(\mu)$ for every singular cardinal $\mu$.

We introduce a new method for controlling $\operatorname{MAD} (\mu)$ for a singular $\mu$ and, among other new results about the structure of $\operatorname{MAD}(\mu)$ for singular $\mu$, settle both problems affirmatively.


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  • 1. B. Balcar and P. Simon. On collections of almost disjoint families, Comment. Math. Univ. Carolinae 29 (1988), no 4, pp. 631-646. MR 90b:03072
  • 2. B. Balcar and P. Simon. Disjoint refinements, in: Handbook of Boolean Algebras, eds. J. D. Monk, R. Bonnet, vol. 2. North-Holland, Amsterdam, 1989.
  • 3. J. E. Baumgartner. Almost-disjoint sets, the dense set problem and the partition calculus, Ann. Math. Logic 9 (1976), no. 4, 401-439. MR 53:5299
  • 4. A. BLASS, Simple cardinal characteristics of the continuum, in: Set theory of the reals, Israel Math. Conf. Proc. 6, Bar-Ilan Univ., Ramat Gan, 1993. (1991) 63-90. MR 94i:03098
  • 5. J. Brendle. The almost disjointness number may have countable cofinality, Trans. Amer. Math. Soc. 355 (2003), no. 7, 2633-2649 (electronic). MR 2004c:03062
  • 6. E. K. VAN DOUWEN, The integers and topology, in: Handbook of set-theoretic topology, 111-167, North-Holland, Amsterdam, 1984. MR 87f:54008
  • 7. P. ERD´´OS, S. HECHLER, On maximal almost-disjoint families over singular cardinals, Colloquia Mathematica Societatis János Bolyai 10, Infinite and finite sets, Keszthely (Hungary) 1973, pp. 597-604. MR 51:12530
  • 8. S. Hechler. Short complete nested sequences in $\beta\mathbb N-\mathbb N$ and small maximal almost disjoint families, Gen. Top. Appl. 2 (1972), 139-149. MR 46:7028
  • 9. M. Gitik and M. Magidor. The singular cardinal hypothesis revisited, Set theory of the continuum (Berkeley, CA, 1989), 243-279, Math. Sci. Res. Inst. Publ., 26, Springer, New York, 1992. MR 95c:03131
  • 10. M. Kojman. Exact upper bounds and their uses in set theory, Ann. Pure Appl. Logic 92 (1998), no. 3, 267-282. MR 2000b:03163
  • 11. E. MILNER, K. PRIKRY, Almost disjoint sets, in: Surveys in combinatorics, (New Cross 1987) pp. 155-172, Cambridge, 1987. MR 88k:04001
  • 12. J. D. MONK, The spectrum of partitions of a Boolean algebra, Arch. Math. Logic 40 (2001), no. 4, 243-254. MR 2002j:03073
  • 13. S. SHELAH, The singular cardinals problem: independence results., Surveys in set theory, 116-134, London Math. Soc. Lecture Note Ser., 87, Cambridge Univ. Press, Cambridge, 1983. MR 87b:03114
  • 14. S. Shelah. Reflecting stationary sets and successors of singular cardinals, Arch. Math. Logic 31 (1991), no. 1, 25-53. MR 93h:03072
  • 15. S. Shelah. Cardinal Arithmetic, Oxford University Press, 1994. MR 96e:03001
  • 16. S. Shelah. Are $\mathfrak a$ and $\mathfrak d$ your cup of tea?, Acta Math., to appear.

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Additional Information

Menachem Kojman
Affiliation: Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva, Israel
Email: kojman@math.bgu.ac.il

Wieslaw Kubis
Affiliation: Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva, Israel – and – Institute of Mathematics, University of Silesia, Katowice, Poland
Email: kubis@math.bgu.ac.il

Saharon Shelah
Affiliation: Institute of Mathematics, Hebrew University of Jerusalem, Israel – and – Department of Mathematics, Rutgers University, New Brunswick, New Jersey
Email: shelah@math.huji.ac.il

DOI: https://doi.org/10.1090/S0002-9939-04-07580-X
Keywords: Almost disjoint family, singular cardinal, bounding number, smooth pcf scales
Received by editor(s): June 10, 2002
Received by editor(s) in revised form: September 10, 2002
Published electronically: June 21, 2004
Additional Notes: The first author’s research partially supported by an Israeli Science Foundation grant no. 177/01
The third author’s research was supported by The Israel Science Foundation, Publication 793.
Communicated by: Carl G. Jockusch, Jr.
Article copyright: © Copyright 2004 American Mathematical Society

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