Simplicial vertex boundaries need not increase below half volume
= Simplicial vertex boundaries need not increase below half volume
In dimension three, the first three vertices are $\varnothing,\{1\},\{2\}$. Their closed <vertex neighbourhood> has seven vertices. Adding $\{3\}$ does not change that <vertex neighbourhood>, so the external <vertex boundary> decreases from four to three even though both family sizes are at most half the cube.