Prove that in any set $S, |S| = 3k$ there exist $2^k$ subsets $V_i$ such that $|V_i| = 0 \mod 3$ and the cardinality of any intersection is $0 \mod 3$ as well.
2026-05-05 06:42:57.1777963377
Simple Graph Set Question
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Divide the elements of $S$ up into $k$ blocks of three. Then how many ways are there to select a collection of blocks? (Hint: you're effectively just selecting a subset of $\{ \text{the $k$ three-sized blocks} \}$.)