Extensions of the Scherk-Kemperman Theorem

摘要

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. {pipe}F{pipe}+{pipe}Γ(v){pipe}-{pipe}Γ-(v)∩F{pipe}.Let A,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: {pipe}AB{pipe}≥{pipe}A{pipe}+{pipe}B{pipe}-{pipe}A∩(cB-1){pipe}, for every c∈AB.