 (iv ) For every sufficiently large n, there exists a set P of n points  such that every plane hop spanner on P has hop stretch factor at least 4. Previously, no lower bound greater than 2 was known.
<meaning-changed>  (iv ) For every sufficiently large n, there exists a set P of n points  such that every plane hop spanner on P has hop stretch factor at least 4. Previously, no lower bound greater than 2 was known.
 (iv ) For every sufficiently large n, there exists a set P of n points on a circle, such that every plane hop spanner on P has hop stretch factor at least 4. Previously, no lower bound greater than 2 was known.