For a set $X$ we endow the set $omega^X$ of all functions from $X$ to $omega$ with the natural partial order $le$ defined by $fle g$ iff $f(x)le g(x)$ for all $xin X$. A function $mu:omega^omegato omega^X$ is called *monotone* if for any $fle g$ in $omega^omega$ we have $mu(f)lemu(g)$.

Question.Is there a monotone function $mu:omega^omegatoomega^{omega_1}$ which iscountably unboundedin the sense that for every countable infinite set $Asubsetomega_1$ and function $finomega^A$ there exist a function $ginomega^omega$ and an infinite set $Bsubseteq A$ such that $f{restriction}_Blemu(g){restriction}_B$?

**Remark.** By the answer of Johannes Schurz to this question, for every monotone function $mu:omega^omegatoomega^{omega_1}$, there exists a countable set $Asubsetomega_1$ and a function $finomega^A$ such that $fnotle mu(g){restriction}A$ for every $ginomega^omega$.