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 | References | Similar Articles | Additional Information

Abstract: A ``-rule" is a sequence of pairwise disjoint sets , each of cardinality and subsets . A subset (a ``real'') follows a rule if for infinitely many , .

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

Call a *bounded* rule if is a -rule for some . Let be the least cardinality of a set of bounded rules with no real following all rules in the set.

We prove the following: and for all . However, in the Laver model, .

An application of is in Section 3: we show that below 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 . The consistency of such a family is still open.

**1.**Jörg Brendle. Evasion and prediction-the Specker phenomenon and Gross spaces.*Forum Math.*, 7(5):513-541, 1995. MR**96i:03042****2.**Martin Goldstern, Rami Grossberg, and Menachem Kojman. Infinite Homogeneous Bipartite Graphs With Unequal Sides.*Discrete Mathematics*, 149:69-82, 1996. MR**97a:05102****3.**Menachem Kojman and Saharon Shelah. Homogeneous families and their automorphism groups.*Journal of the London Mathematical Society*, 52:303-317, 1995.**4.**R. Laver. On the consistency of Borel's conjecture.*Acta Math.*, 137:151-169, 1976. MR**54:10019****5.**J. K. Truss. Embeddings of infinite permutation groups. In*Proceedings of groups - St Andrews 1985*, volume 121 of*London Math. Soc. Lecture Series*, pages 335-351. Cambridge University Press, 1986. MR**89d:20002****6.**J.E. Vaughan. Small uncountable cardinals in topology. In*Open problems in topology*, pages 217-218. Elsvier Science Publishers, B.V. North Holland, 1990. ed. van Mill, J. and Reed, G.M.

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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