Rules and reals

Abstract:

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