  Given an alphabet $S$, we consider the size of the subsets of the full
sequence space $S^{\rm {\bf Z}}$ determined by the additional restriction that
$x_i\not=x_{i+f(n)},\ i\in {\rm {\bf Z}},\ n\in {\rm {\bf N}}.$ Here $f$ is a
positive, strictly increasing function. We review an other, graph theoretic,
formulation and then the known results covering various combinations of $f$ and
the alphabet size. In the second part of the paper we turn to the fine
structure of the allowed sequences in the particular case where $f$ is a
suitable polynomial. The generation of sequences leads naturally to consider
the problem of their maximal length, which turns out highly random
asymptotically in the alphabet size.
