Eventual domination ($f{\le }^{\ast }g$)

For functions $f,g:\omega \to \omega$, write $f{\le }^{\ast }g$ when $f(n)\le g(n)$ for all sufficiently large $n$. A dominating family is cofinal for this relation.

The least size of a family in ${\omega }^{\omega }$ having no common bound for eventual domination. Every countable family $(f_i)$ has the bound $g(n)=1+{\max }_{i\le n}{f}_i(n)$, so ${\aleph }_1\le \mathfrak{b}\le 2^{{\aleph }_0}$.

A family $C\subseteq {\omega }^{\omega }$ such that every $f\in {\omega }^{\omega }$ is eventually dominated by some $g\in C$.