Rules and reals

Abstract: "A 'k-rule' is a sequence [superscript ->]A = ((A[subscript n], B[subscript n]) : n < [omega]) of pairwise disjoint sets B[subscript n], each of cardinality [< or =] k and subsets A[subscript n] [subset] B[subscript n]. A subset X [subset omega] (a 'real') follows a rule [superscript ->]A if for infinitely many n [element] w, X [intersect] B[subscript n] = A[subscript n]. There are obvious cardinal invariants resulting from this definition: the least number of reals needed to follow all k-rules, s[subscript k], and the least number of k-rules without a real following all of them, t[subscript k]. Call [superscript ->]A a bounded rule if [superscript ->]A is a k-rule for some k. Let t[subscript infinity] be the least cardinality of a set of bounded rules with no real following all rules in the set. We prove the following: t[subscript infinity [> or =] max(cov(K), cov(L)) and t = tΓéü [> or =] tΓéé = t[subscript k] for all k [> or =] 2. However, in the Laver model, tΓéé < b = tΓéü. An application of t[subscript infinity] is in Section 3: we show that below t[subscript infinity] one can find proper extensions of dense independent families which preserve a pre-assigned group of automorphism. 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."