  A subset $U$ of a set $S$ with a binary operation is called {\it avoidable}
if $S$ can be partitioned into two subsets $A$ and $B$ such that no element of
$U$ can be written as a product of two distinct elements of $A$ or as the
product of two distinct elements of $B$. The avoidable sets of the bicyclic
inverse semigroup are classified.
