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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?
Community
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Martin Sleziak
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    Sorry, I clicked the wrong button: If ${f_i : i \in I}$ is almost disjoint then $f_i \neq f_j$ for $i \neq j$, so $|I| \leq |\omega^\omega| =\mathfrak{c}$. – Martin Apr 09 '13 at 06:43
  • Thanks @Martin, I've edited my question accordingly. (And I am really ashamed that I have missed this.) – Martin Sleziak Apr 09 '13 at 06:45

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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

  • Yi Zhang, On a class of m.a.d. families, J. Symbolic Logic 64 (1999), no.2, pp.737–746, MR1777782.
  • Dilip Raghavan, Maximal almost disjoint families of functions, Fund. Math. 204 (2009), no.3, pp.241–282, MR2520154.
user642796
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    But MAD families sounds so much cooler. One may even prove a theorem on the behavior of MAD families under certain changes (e.g. passing to elementary submodel) and name it after mad families from TV (e.g. Simpsons, Griffins, Addams)... – Asaf Karagila Apr 09 '13 at 07:47
  • @Asaf: But are MAD families as cool as very MAD families, or perhaps mob families? Ahhh... set theorists have become quite cynical since Mathias! – user642796 Apr 09 '13 at 08:42
  • We should talk about NUTBAG "Nowhere Ultra-Tame Borel Almost Generic" sets, which are sets which are really crazy. – Asaf Karagila Apr 09 '13 at 08:54