Let us call a system $\{f_i; i\in I\}$ of functions $f_i\colon\omega\to\omega$ almost disjoint if the set $\{n\in\omega; f_i(n)=f_j(n)\}$ is finite whenever $i\ne j$. (I am not entirely sure whether this is a standard terminology for such system of functions, but the same name was used in a different question on this site.)
Let $\mathfrak m$ denotes the least cardinality of a maximal almost disjoint system of functions in $\omega^\omega$. One part of my answer here in fact shows $\aleph_1 \le \mathfrak m$. A more general version of this lower bound was shown in this question.
My guess is that we also have $\mathfrak m\le \mathfrak c$, although I don't have a proof for this, since the cardinality of $\omega^\omega$ is $\mathfrak c$. (This was pointed out in a comment below; I have somehow missed this easy estimate. I'll try to get away with the excuse that I was posting this relatively early in the morning. :-)
- Was such cardinal studied? And if yes, what is it called?
Is it true that $\mathfrak m\le \mathfrak c$?- Is it perhaps equal to some of other small uncountable cardinals?
Almost disjoint families of functions $\omega \to \omega$ are sometimes called (pairwise) eventually different families. I have seen the notation $\mathfrak{a}_{\mathfrak{e}}$ used for the corresponding cardinal invariant, for example in