Is this a well-known NP-hard problem? Let $R = \{1, \ldots, n\}$ and $S = \{S_1, \ldots, S_m\}$ a collection of subsets of $R$ such that $R = \bigcup_{i = 1}^m S_i$ and, for $n &gt; 3$, $$3 \leq \vert S_i \vert \leq 4 \, , \enspace i \in \{1, \ldots, m\} \, .$$

Then, I want to know the subset—or subsets, since there may be more than one valid solution—$T$ with minimum cardinality such that every $S_i$ has at least one element in $T$. I suspect this is an NP-hard problem (or NP-complete in its decision version), but I don't know if it's one that has a name.

As an example, consider $R = \{1, 2, 3, 4, 5\}$ and $S = \{S_1, \ldots, S_9\}$, where


$S_1 = \{1, 2, 3\} \, , \enspace S_4 = \{1, 4, 5\} \, , \enspace S_7 = \{1, 2, 3, 4\} \, ,$
$S_2 = \{1, 2, 4\} \, , \enspace S_5 = \{2, 3, 5\} \, , \enspace S_8 = \{1, 3, 4, 5\} \, ,$
$S_3 = \{1, 2, 5\} \, , \enspace S_6 = \{3, 4, 5\} \, , \enspace S_9 = \{2, 3, 4, 5\} \, .$


Here, the solutions are $T = \{\{1, 3\}, \{1, 5\}, \{2, 4\}, \{2, 5\}\}$. (I'd be happy even if I knew just one of them.)

Note that I'm not asking for an algorithm to solve the problem. I just want to know where this is or reduces to a well-known problem.
3-Hitting Set problem is known in parameterized complexity theory. The requirement $\cup S_i=R$ can always be assumed without loss of generality. See e.g. An efficient fixed-parameter algorithm for 3-Hitting Set. According to this link it is NP-hard in its usual (not parameterized) form. Proving NP-completeness of your problem we give reduction FROM 3-Hitting Set to your problem not vica versa. Therefore your problem is NP-complete (in its decision form).