  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.
