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.

For $f,g:\kappa \to \kappa$, write $f{<}^{\ast }g$ when there is $\gamma <\kappa$ such that $f(\delta )<g(\delta )$ whenever $\gamma <\delta <\kappa$. For an infinite regular cardinal, suprema of fewer than $\kappa$ ordinals below $\kappa$ remain below it, permitting the long chain under eventual domination construction.

For every infinite regular cardinal $\kappa$, there is a strictly eventually increasing sequence of ${\kappa }^{+}$ functions $\kappa \to \kappa$. At stage $0<\zeta <{\kappa }^{+}$ enumerate predecessors by $e:\kappa \to \zeta$ and put $f_{\zeta }(\delta )={\sup }_{\eta <\delta }(f_{e(\eta )}(\delta )+1)$. Regularity bounds each value below $\kappa$, while any predecessor's index is eventually included in the supremum.

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 $g\in C$.