6 Ju l 1 99 7 RULES AND REALS

A " k-rule " is a sequence A = ((A n , B n) : n < ω) of pairwise disjoint sets B n , each of cardinality ≤ k and subsets A n ⊆ B n. A subset X ⊆ ω (a " real ") follows a rule A if for infinitely many n ∈ ω, X ∩ B n = A n. There are obvious cardinal invariants resulting from this definition: the least number of reals needed to follow all k-rules, s k , and the least number of k-rules without a real following all of them, r k. Call A a bounded rule if A is a k-rule for some k. Let r ∞ be the least cardinality of a set of bounded rules with no real following all rules in the set. We prove the following: r ∞ ≥ max(cov(K), cov(L)) and r = r 1 ≥ r 2 = r k for all k ≥ 2. However, in the Laver model, r 2 < b = r 1. An application of r ∞ is in Section 3: we show that below r ∞ 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.