Extensions of the Scherck-Kemperman Theorem

TL;DR

This paper extends the Scherck-Kemperman Theorem to reflexive relations with transitive automorphism groups, providing a lower bound on the size of the image of a finite set, with applications to Cayley graphs.

Contribution

It generalizes the Scherck-Kemperman Theorem to broader relational structures with symmetry properties.

Findings

Established a lower bound for the size of the image of a finite set in reflexive relations.

Extended the Scherck-Kemperman Theorem to Cayley graphs and group subsets.

Provided a new inequality involving set images and intersections in symmetric relations.

Abstract

Let $\Gamma =(V,E)$ be a reflexive relation with a transitive automorphisms group. Let $v\in V$ and let $F$ be a finite subset of $V$ with $v\in F.$ We prove that the size of $\Gamma (F)$ (the image of $F$) is at least $$ |F|+ |\Gamma (v)|-|\Gamma ^- (v)\cap F|.$$ Let $A,B$ be finite subsets of a group $G.$ Applied to Cayley graphs, our result reduces to following extension of the Scherk-Kemperman Theorem, proved by Kemperman: $$|AB|\ge |A|+|B|-|A\cap (cB^{-1})|,$$ for every $c\in AB.$