## Abstract

A "k-rule" is a sequence A→ = ((A_{n},B_{n}) : n < ℕ) of pairwise disjoint sets B_{n}, each of cardinality ≤ k and subsets A_{n} ⊆ B_{n}. A subset X ⊆ ℕ (a "real") follows a rule A→ if for infinitely many n ∈ ℕ, X ∩ B_{n} = A_{n}. Two obvious cardinal invariants arise from this definition: the least number of reals needed to follow all k-rules, s_{k}, and the least number of k-rules with no real that follows all of them, τ_{k}. Call A→ a bounded rule if A→ is a k-rule for some k. Let τ∞ be the least cardinality of a set of bounded rules with no real following all rules in the set. We prove the following: τ∞ ≥ max(cov(K),cov(L)) and τ = τ_{1} ≥ τ_{2} = τ_{k} for all k ≥ 2. However, in the Laver model, τ_{2} < b = τ_{1}. An application of τ_{∞} is in Section 3: we show that below τ_{∞} 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.

Original language | English |
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Pages (from-to) | 1517-1521 |

Number of pages | 5 |

Journal | Proceedings of the American Mathematical Society |

Volume | 127 |

Issue number | 5 |

DOIs | |

State | Published - 1 Jan 1999 |

## Keywords

- Cardinal invariants of the continuum