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I am writing an essay for which I need to prove that sufficiently many graphs of a certain type exist. Is it true that for any set of sets (or set of cardinals) $S$ a countable subset $C$ exists such that

$$\forall s \in S \exists c\in C[c>s]$$

if we speak about cardinals or

$$\forall s \in S \exists c\in C[|c|>|s|]$$

when speaking about sets of sets? does this require well ordering or any other kind of explanation?

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No, this is not always true. For example $S$ could be the set of cardinals less than $\aleph_{\omega_1}$. Your $C$ would be a set of alephs whose indices have $\omega_1$ as their least upper bound, but that's not possible for a countable $C$ because $\omega_1$ has cofinality $\omega_1$.

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