Stellar subdivision injects a complex's minimal non-faces into those of its refinement

Let $K$ be a finite simplicial complex, $\sigma$ a nonempty face of $K$, $a$ a new vertex used by no face of $K$, and $K'$ the stellar subdivision of $K$ at $(\sigma,a)$. Then there is an injective map $\varphi$ from $K$'s minimal non-faces to $K'$'s minimal non-faces, given by $\varphi(M)=M$ if $\sigma\not\subseteq M$ and $\varphi(M)=(M\setminus\sigma)\cup\{a\}$ otherwise.

So a single stellar subdivision step never decreases the number of witnessed minimal non-faces — the quantity that must shrink to zero for a complex to become flag. This depends on the corrected IsMinimalNonface in Hirsch_simplicial_complex_flag (restricted to $K$'s own vertex support); without that restriction an unused vertex $a$ gives a vacuous minimal non-face $\{a\}$ of $K$ whose image $\varphi(\{a\})=\{a\}$ becomes a genuine face of $K'$, falsifying injectivity.