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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Rules and reals


Authors: Martin Goldstern and Menachem Kojman
Journal: Proc. Amer. Math. Soc. 127 (1999), 1517-1524
MSC (1991): Primary 03E35; Secondary 03E50, 20B27
DOI: https://doi.org/10.1090/S0002-9939-99-04635-3
Published electronically: January 29, 1999
MathSciNet review: 1473670
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Abstract: A ``$k$-rule" is a sequence $\vec A=((A_n,B_n): n<\mathbb N)$ of pairwise disjoint sets $B_n$, each of cardinality $\le k$ and subsets $A_n\subseteq B_n$. A subset $X\subseteq \mathbb N$ (a ``real'') follows a rule $\vec A$ if for infinitely many $n\in \mathbb N$, $X\cap B_n=A_n$.

Two obvious cardinal invariants arise from this definition: the least number of reals needed to follow all $k$-rules, $\mathfrak s_k$, and the least number of $k$-rules with no real that follows all of them, $\mathfrak r_k$.

Call $\vec A$ a bounded rule if $\vec A$ is a $k$-rule for some $k$. Let $\mathfrak r_\infty$ be the least cardinality of a set of bounded rules with no real following all rules in the set.

We prove the following: $\mathfrak r_\infty\ge\max(\operatorname{cov}(\mathbb K),\operatorname{cov}(\mathbb L))$ and $\mathfrak r=\mathfrak r_1\ge \mathfrak r_2=\mathfrak r_k$ for all $k\ge 2$. However, in the Laver model, $\mathfrak r_2<\mathfrak b=\mathfrak r_1$.

An application of $\mathfrak r_\infty$ is in Section 3: we show that below $\mathfrak r_\infty$ one can find proper extensions of dense independent families which preserve a pre-assigned group of automorphisms. The original motivation for discovering rules was an attempt to construct a maximal homogeneous family over $\omega $. The consistency of such a family is still open.


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Additional Information

Martin Goldstern
Affiliation: Institut für Algebra, Technische Universität, Wiedner Hauptstraße 8–10/118.2, A-1040 Wien, Austria
Email: Martin.Goldstern@tuwien.ac.at

Menachem Kojman
Affiliation: Department of Mathematics, Ben–Gurion University of the Negev, POB 653. Beer-Sheva 84105, Israel
Address at time of publication: Department of Mathematics, Carnegie Mellon University, 5000 Forbes Ave, Pittsburgh, Pennsylvania 15213
Email: kojman@math.bgu.ac.il

DOI: https://doi.org/10.1090/S0002-9939-99-04635-3
Keywords: Cardinal invariants of the continuum
Received by editor(s): February 4, 1997
Received by editor(s) in revised form: July 16, 1997
Published electronically: January 29, 1999
Additional Notes: The first author is supported by an Erwin Schrödinger fellowship from the Austrian Science Foundation (FWF)
The second author was partially supported by NSF grant no. 9622579.
Communicated by: Carl Jockusch
Article copyright: © Copyright 1999 American Mathematical Society