Extensions of the Scherk–Kemperman Theorem

Let Γ=(V,E)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)Γ(F) (the image of F) is at least|F|+|Γ(v)|−|Γ−(v)∩F|.|F|+|Γ(v)|−|Γ−(v)∩F|.Let A,BA,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:|AB|⩾|A|+|B|−|A∩(cB−1)|,|AB|⩾|A|+|B|−|A∩(cB−1)|, for every c∈ABc∈AB.