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Rules and Reals

Proceedings of the American Mathematical Society / v.127 no.5. 1999, pp.1517-1524

Author : Goldstern, Martin ; Kojman, Menachem

Abstract : A k-rule is a sequence $\vec{A}$ = ((Alt;LAT ((A $_{n}$ , B $_{n}$ ): n ${\Bbb N}$ ) of pairwise disjoint sets B $_{n}$ , each of cardinality $\leq $ k and subsets A $_{n}\subseteq $ B $_{n}$ . A subset X $\subseteq {\Bbb N}$ (a real) follows a rule $\vec{A}$ if for infinitely many n $\in {\Bbb N}$ , X $\cap $ B $_{n}$ = Alt;LA A $_{n}$ . Two obvious cardinal invariants arise from this definition: the least number of reals needed to follow all k-rules, $\germ{s}_{k}$ , and the least number of k-rules with no real that follows all of them, $\germ{r}_{k}$ . Call $\vec{A}$ a bounded rule if $\vec{A}$ is a k-rule for some k. Let $\germ{r}_{\infty}$ be the least cardinality of a set of bounded rules with no real following all rules in the set. We prove the following: $\germ{r}_{\infty}\geq $ max(cov( ${\Bbb K}$ ), cov( ${\Bbb L}$ )) and $\germ{r}=\germ{r}_{1}\geq \germ{r}_{2}=\germ{r}_{k}$ for all k $\geq 2$ . However, in the Laver model, $\germ{r}_{2} . An application of $\germ{r}_{\infty}$ is in Section 3: we show that below $\germ{r}_{\infty}$ 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.

Keyword : Primary 03E35 . Secondary 03E50, 20B27 . Cardinal invariants of the continuum

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