The splitting and reaping numbers

In ZFC, with $[\omega {]}^{\omega }$ the infinite subsets of $\omega$ (Almost inclusion, pseudointersections and towers) and $\mathfrak{c}=2^{{\mathrm{\aleph }}_0}$ (Cardinal sum $\kappa \oplus \lambda$, product $\kappa \otimes \lambda$ and exponentiation ${\kappa }^{\lambda }$, and why they are written apart from the ordinal operations):

Splitting. For $X,Y\subseteq \omega$, say that $X$ splits $Y$ when both $Y\cap X$ and $Y\setminus X$ are infinite. A family $\mathcal{S}\subseteq [\omega {]}^{\omega }$ is a splitting family when every $Y\in [\omega {]}^{\omega }$ is split by some member of $\mathcal{S}$. The splitting number is

$s:=\min \{\mid \mathcal{S}\mid :\mathcal{S}\subseteq [\omega {]}^{\omega }\text{ is a splitting family}\}\mathrm{.}$

Reaping. A family $\mathcal{R}\subseteq [\omega {]}^{\omega }$ is unreaped when no single set $X\subseteq \omega$ splits every member of $\mathcal{R}$; the negation, "$\mathcal{R}$ is reaped by $X$", thus means that $X$ splits each $Y\in \mathcal{R}$. The reaping number is

$r:=\min \{\mid \mathcal{R}\mid :\mathcal{R}\subseteq [\omega {]}^{\omega }\text{ is unreaped}\}\mathrm{.}$

Both minima exist and are cardinals. Candidates for $s$ are subsets of $[\omega {]}^{\omega }$, and the minimum is attained: $[\omega {]}^{\omega }$ itself is a splitting family, since $Y\in [\omega {]}^{\omega }$ has an increasing enumeration $y_0<y_1<\cdots$ and the even part $\{y_{2n}:n\in \mathbb{N}\}$ is an infinite subset of $Y$ whose complement in $Y$ is the infinite set of odd-indexed elements. Candidates for $r$ are again subsets of $[\omega {]}^{\omega }$, and $[\omega {]}^{\omega }$ itself is unreaped: given $X\subseteq \omega$, either $\omega \setminus X$ is infinite, in which case the member $\omega \setminus X$ of $[\omega {]}^{\omega }$ meets $X$ in the empty set and so is not split by $X$, or $\omega \setminus X$ is finite, in which case the member $X$ of $[\omega {]}^{\omega }$ satisfies $X\setminus X=\mathrm{\varnothing }$ and so is not split by $X$ either. In both cases some member of $[\omega {]}^{\omega }$ is not split by $X$, so $r\le \mid [\omega {]}^{\omega }\mid \le \mathfrak{c}$ and $s\le \mathfrak{c}$; the cardinalities come from the Axiom of Choice (The Axiom of Choice, A set equinumerous with some ordinal has a least such ordinal, that ordinal is a cardinal, and equinumerous sets get the same one; no choice principle is used).

Conventions. Splitting is asymmetric: $X$ splits $Y$ is a statement about $Y$'s two parts, and it implies $Y$ is infinite but imposes no infinitude condition on $X$ beyond $Y\cap X$ being infinite; splitting families are, however, customarily taken inside $[\omega {]}^{\omega }$, as above. The comparison of $s$ and $r$ with $b$, $d$ and $\mathfrak{c}$ is Splitting and reaping comparisons with b and d; the elementary lower bounds ${\mathrm{\aleph }}_1\le s$ and ${\mathrm{\aleph }}_1\le b\le r$ are proved there, and the upper bounds are the ones just displayed.

Monk attributes the splitting number and the diagonal lower bound $\omega <s$ to Blass, and the inequality $b\le r$ is his Proposition 27; the reaping number is the least size of a family that no single set splits, which is the formulation used consistently below. The names come from the dual picture: $\mathcal{R}$ is unreaped by $X$ when $X$ fails to split some member, so in a reaping family every candidate splitter fails on at least one member.