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Media type:
E-Article
Title:
Rules and Reals
Contributor:
Goldstern, Martin;
Kojman, Menachem
imprint:
American Mathematical Society, 1999
Published in:Proceedings of the American Mathematical Society
Language:
English
ISSN:
0002-9939;
1088-6826
Origination:
Footnote:
Description:
<p>A "k-rule" is a sequence<tex-math>$\vec{A}$</tex-math>= ((A<sub>n</sub>, B<sub>n</sub>): n < N) of pairwise disjoint sets B<sub>n</sub>, each of cardinality ≤ k and subsets A<tex-math>$_{n}\subseteq $</tex-math>B<sub>n</sub>. A subset X<tex-math>$\subseteq {\Bbb N}$</tex-math>(a "real") follows a rule<tex-math>$\vec{A}$</tex-math>if for infinitely many n ∈ N, X ∩ B<sub>n</sub>= A<sub>n</sub>. Two obvious cardinal invariants arise from this definition: the least number of reals needed to follow all k-rules, s<sub>k</sub>, and the least number of k-rules with no real that follows all of them, r<sub>k</sub>. Call<tex-math>$\vec{A}$</tex-math>a bounded rule if<tex-math>$\vec{A}$</tex-math>is a k-rule for some k. Let r<sub>∞</sub>be the least cardinality of a set of bounded rules with no real following all rules in the set. We prove the following: r<sub>∞</sub>≥ max(cov( K), cov( L)) and r=r<sub>1</sub>≥ r<sub>2</sub>=r<sub>k</sub>for all k ≥ 2. However, in the Laver model,<tex-math>$\germ{r}_{2}<\germ{b}=\germ{r}_{1}$</tex-math>. An application of r<sub>∞</sub>is in Section 3: we show that below r<sub>∞</sub>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 ω . The consistency of such a family is still open.</p>