= Coordinate-shift shadow containment
{title2=$\partial C_{ij}\mathcal F\subseteq C_{ij}\partial\mathcal F$}
For a <uniform set family>, the <lower shadow> of its shifted <set family> is contained in the shifted <lower shadow>. Check the witness in the four patterns of membership in coordinates $i,j$. A surviving shadow member containing only $j$ must have both old shadow partners; a member containing only $i$ needs either old partner. Taking complements exchanges $C_{ij}$ with $C_{ji}$ and gives the same containment for the <upper shadow>. Thus <coordinate shifts of a set family> cannot increase either shadow or their disjoint union, the <external vertex boundary> of a <uniform set family>.
Back to article page
