Extensions of the Scherk–Kemperman Theorem

Let Γ = ( V , E ) be a reflexive relation with a transitive automorphism group. Let F be a finite subset of V containing a fixed element v. We prove that the size of Γ ( F ) (the image of F) is at least | F | + | Γ ( v ) | − | Γ − ( v ) ∩ F | . , B be finite subsets of a group G. Applied to Cayley graphs, our result reduces to the following extension of the Scherk–Kemperman Theorem, proved by Kemperman: | A B | ⩾ | A | + | B | − | A ∩ ( c B − 1 ) | , for every c ∈ A B .