 This is the first construction of a 2-hop spanner with a subquadratic number of edges. (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.
<clarity>  This is the first construction of a 2-hop spanner with a subquadratic number of edges. (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.
 This is the first construction of a 2-hop spanners. (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.