Extensions of the Moser-Scherck-Kemperman-Wehn Theorem

Let $\Gamma =(V,E)$ be a reflexive relation having a transitive group of automorphisms and let $v\in V.$ Let $F$ be a subset of $V$ with $F\cap \Gamma ^-(v)=\{v\}$. (i) If $F$ is finite, then $| \Gamma (F)\setminus F|\ge |\Gamma (v)|-1.$ (ii) If $F$ is cofinite, then $| \Gamma (F)\setminus F|\ge |\Gamma ^- (v)|-1.$ In particular, let $G$ be group, $B$ be a finite subset of $G$ and let $F$ be a finite or a cofinite subset of $G$ such that $F\cap B^{-1}=\{1\}$. Then $| (FB)\setminus F|\ge |B|-1.$ The last result (for $F$ finite), is famous Moser-Scherck-Kemperman-Wehn Theorem. Its extension to cofinite subsets seems new. We give also few applications.