On iterated image size for point-symmetric relations

Let $\Gamma =(V,E)$ be a point-symmetric reflexive relation and let $v\in V$ such that $|\Gamma (v)|$ is finite (and hence $|\Gamma (x)|$ is finite for all $x$, by the transitive action of the group of automorphisms). Let $j\in \N$ be an integer such that $\Gamma ^j(v)\cap \Gamma ^{-}(v)=\{v\}$. Our main result states that $$|\Gamma ^{j} (v)|\ge | \Gamma ^{j-1} (v)| + |\Gamma (v)|-1.$$ As an application we have $|\Gamma ^{j} (v)| \ge 1+(|\Gamma (v)|-1)j.$ The last result confirms a recent conjecture of Seymour in the case of vertex-symmetric graphs. Also it gives a short proof for the validity of the Caccetta-H\"aggkvist conjecture for vertex-symmetric graphs and generalizes an additive result of Shepherdson.