Rules and reals
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- by Martin Goldstern and Menachem Kojman PDF
- Proc. Amer. Math. Soc. 127 (1999), 1517-1524 Request permission
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.References
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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
- 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
- © Copyright 1999 American Mathematical Society
- 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
- MathSciNet review: 1473670