A rule is a sequence \(\vec A=((A_n,B_n):n\in\omega)\) where the sets \(B_n\) are finite and pairwise disjoint, and \(A_n\subseteq B_n\subseteq \omega\) for all \(n\). The rule \(\vec A\) is a \(k\)-rule (for \(k\in\omega\)) if \(| B_n| \leq k\) for all \(n\); the rule is a bounded rule if it is a \(k\)-rule for some \(k\). An infinite subset \(X\) of \(\omega\) is said to follow the rule \(\vec A\) if \(X\cap B_n=A_n\) for infinitely many \(n\).
The authors introduce the following cardinal invariants related to these rules: \(\mathfrak s_k\), the least cardinality of a family \(\mathcal S\) such that every \(k\)-rule is followed by some \(X\) in \(\mathcal S\), and \(\mathfrak r_k\) (or \(\mathfrak r_\infty\)), the least cardinality of a family \(\mathcal R\) of \(k\)-rules (or of bounded rules) such that there is no \(X\subseteq\omega\) which follows all the rules in \(\mathcal R\).
The paper compares these invariants among themselves and to other well-known invariants of the continuum: for example, \(\mathfrak r=\mathfrak r_1 \geq \mathfrak r_2=\mathfrak r_k\) for all \(k\geq 2\); \(\mathfrak r_\infty \geq{}\) the smallest number of first category sets (or measure zero sets) needed to cover the real line. In the Laver model, \(\mathfrak r_2<\mathfrak b=\mathfrak r_1\). In the last section, the authors show that below \(\mathfrak r_\infty\) one can properly extend a dense independent family of subsets of \(\omega\) preserving a prescribed group of automorphisms.