Given a set of sets: $S = \{~\{1, 2, 3\}, \{2, 3, 4\}, \{1, 3, 4\}~\}$, I would like to find the largest common subset of $S$. If $S$ does not have a subset across all elements of $S$, I would like to find the largest common subset across the largest subset of $S$.
For example:
$S = \{~\{1, 5, 4\}, \{2, 3, 4\}, \{1, 3, 4\}~\}$ has $\{4\}$ in common across all subsets.
$T = \{~\{1, 2, 6\}, \{2, 3, 4\}, \{1, 3, 4\}~\}$ has $\{3, 4\}$ in common across the subset $\{T_1, T_2\}$