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Is there a notation for the collection of all sets with cardinality $k$, for some integer $k$? This collection is not a set, but it is still a useful collection. For example, suppose this collection is denoted by $\mathcal{S}[k]$. Then, given a set-family $H$, we can write:

$$H\cap \mathcal{S}[k]$$

to denote the elements of $H$ with cardinality $k$.

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    $\begingroup$ Not the answer to your question, but for the set of all subsets of $H$ having cardinality $k$, I see to recall I've seen notation something like $[H]^k$. Unfortunately it's impossible to google. $\endgroup$
    – Nate Eldredge
    Apr 25, 2017 at 15:09
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    $\begingroup$ @NateEldredge This is the standard notation. I'm pretty sure that one can find this is Kunen's or Jech's books on set theory. Also, since we usually denote the universe by $V$ or $U$, I think one could write $[V]^k$ or $[U]^{k}$ to denote the desired class. $\endgroup$
    – Renan Mezabarba
    Apr 25, 2017 at 15:12
  • $\begingroup$ This is the class of $k$-subsets of the set universe $V$. It can also be written as $\binom{V}{k}$. $\endgroup$
    – user76284
    Oct 27, 2019 at 20:23

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Generally, in set theory we use $[X]^\kappa$ to denote the set of subsets of $X$ of size exactly $\kappa$. Here $\kappa$ can be an integer, or a transfinite cardinal.

If I had to choose a notation which would be recognizable to a set theorist, I'd use $[V]^\kappa$, where $V$ is the set theoretic universe. You can use other letters if $V$ has a different "usual interpretation" in your context.

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