  A partition of the positive integers into sets $A$ and $B$ {\em avoids} a set
$S\subset\N$ if no two distinct elements in the same part have a sum in $S$. If
the partition is unique, $S$ is {\em uniquely avoidable.} For any irrational
$\alpha>1$, Chow and Long constructed a partition which avoids the numerators
of all convergents to $\alpha$, and conjectured that the set $S_\alpha$ which
this partition avoided was uniquely avoidable. We prove that the set of
numerators of convergents is uniquely avoidable if and only if the continued
fraction for $\alpha$ has infinitely many partial quotients equal to 1. We also
construct the set $S_\alpha$ and show that it is always uniquely avoidable.
