Rules and reals

Abstract

Listen

Translate article

A “ k k -rule" is a sequence A → = ( ( A n , B n ) : n > N ) \vec A=((A_n,B_n): n>\mathbb N) of pairwise disjoint sets B n B_n , each of cardinality ≤ k \le k and subsets A n ⊆ B n A_n\subseteq B_n . A subset X ⊆ N X\subseteq \mathbb N (a “real”) follows a rule A → \vec A if for infinitely many n ∈ N n\in \mathbb N , X ∩ B n = A n X\cap B_n=A_n . Two obvious cardinal invariants arise from this definition: the least number of reals needed to follow all k k -rules, s k \mathfrak {s}_k , and the least number of k k -rules with no real that follows all of them, r k \mathfrak {r}_k . Call A → \vec A a bounded rule if A → \vec A is a k k -rule for some k k . Let r ∞ \mathfrak {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: r ∞ ≥ max ( cov ⁡ ( K ) , cov ⁡ ( L ) ) \mathfrak {r}_\infty \ge \max (\operatorname {cov}(\mathbb {K}),\operatorname {cov}(\mathbb {L})) and r = r 1 ≥ r 2 = r k \mathfrak {r}=\mathfrak {r}_1\ge \mathfrak {r}_2=\mathfrak {r}_k for all k ≥ 2 k\ge 2 . However, in the Laver model, r 2 > b = r 1 \mathfrak {r}_2>\mathfrak {b}=\mathfrak {r}_1 . An application of r ∞ \mathfrak {r}_\infty is in Section 3: we show that below r ∞ \mathfrak {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 ω \omega . The consistency of such a family is still open.