= Solution
\b[False.] The <simplicial vertex boundaries need not increase below half volume>. In $Q_3$,
$$
C(3)=\{\varnothing,\{1\},\{2\}\},\qquad
C(4)=C(3)\cup\{\{3\}\}.
$$
Both <closed graph neighbourhoods> consist of all vertices except $\{1,2,3\}$, so both have size seven. Their external <vertex boundaries> therefore have sizes
$$
\boxed{|\partial C(3)|=4>3=|\partial C(4)|},
$$
although $3<4=2^{3-1}$. Adding a vertex already in the neighbourhood can reduce the external <vertex boundary> without enlarging the neighbourhood.
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