Let $\mathcal{P}(\{0,\cdots,7\})$ denote all subsets of $\{0,\cdots,7\}$.

For any function $f: \mathcal{P}(\{0,\cdots,7\})\rightarrow\{0,1\}$, there exists $0\leq k\leq 3$, three mutually disjoint sets $A,B,C\subseteq \{0,\cdots,7\}$ such that
$\min A = 2k, \min B = 2k+1,$ and $$f(C) = f(A\cup C) = f(B\cup C) = f(A\cup B\cup C).$$