  Let $\Gamma =(V,E)$ be a point-transitive reflexive relation. Let $v\in V$
and put $r=|\Gamma (v)|.$ Also assume $\Gamma ^j(v)\cap \Gamma ^{-}(v)=\{v\}$.
Then $$ |\Gamma ^{j} (v)\setminus \Gamma ^{j-1} (v)| \ge r-1.$$
  In particular we have $ |\Gamma ^{j} (v)| \ge 1+(r-1)j.$ The last result
confirms a recent conjecture of Seymour in the case vertex-transitive graphs.
Also it gives a short proof for the validity of the Caccetta-H\"aggkvist
conjecture for vertex-transitive graphs and generalizes an additive result of
Shepherdson.
