Of course if formula_9 this implies that formula_42, but Hechler has shown that it is also consistent to have formula_21 strictly less than formula_22. We denote by formula_48 the set of all infinite subsets of formula_18. For any formula_50, we say that formula_51 "splits" formula_52 if both formula_53 and formula_54 are infinite. The "splitting number" formula_46 is the least cardinality of a subset formula_56 of formula_48 such that for all formula_58, there is some formula_59 such that formula_51 splits formula_52. That is, formula_62  The "reaping number" formula_47 is the least cardinality of a subset formula_64 of formula_48 such that no element formula_51 of formula_48 splits every element of formula_64. That is, formula_69  The ultrafilter number formula_70 is defined to be the least cardinality of a filter base of a non-principal ultrafilter on formula_18. Kunen gave a model of set theory in which formula_73 but formula_74, and using a countable support iteration of Sacks forcings, Baumgartner and Laver constructed a model in which formula_73 and formula_76. Two subsets formula_78 and formula_79 of formula_18 are said to be "[almost disjoint]" if formula_81 is finite, and a family of subsets of formula_18 is said to be almost disjoint if its members are pairwise almost disjoint. A "maximal almost disjoint" (mad) family of subsets of formula_18 is thus an almost disjoint family formula_84 such that for every subset formula_85 of formula_18 not in formula_84, there is a set formula_88 such that formula_78 and formula_85 are not almost disjoint (that is, their intersection is infinite). The almost disjointness number formula_77 is the least cardinality of an infinite maximal almost disjoint family. A basic result is that formula_92; Shelah showed that it is consistent to have the strict inequality formula_93. A well known diagram of cardinal characteristics is Cichoń's diagram, showing all pair-wise relations provable in ZFC between 10 cardinal characteristics.