Rules and reals

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 = τ₁ ≥ τ₂ = τ_k for all k ≥ 2. However, in the Laver model, τ₂ < b = τ₁. 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.