Extensions of the Moser-Scherck-Kemperman-Wehn Theorem

Let Γ = (V,E) be a reflexive relation having a transitive group of automorphisms and let v ∈ V. Let F be a subset of V with F ∩Γ−(v) = {v}. (i) If F is finite, then |Γ(F ) \ F | ≥ |Γ(v)| − 1. (ii) If F is cofinite, then |Γ(F ) \ F | ≥ |Γ−(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 ∩ B = {1}. Then |(FB) \ F | ≥ |B| − 1. The last result (for F finite), is famous MoserScherck-Kemperman-Wehn Theorem. Its extension to cofinite subsets seems new. We give also few applications.