  Harary's conjecture $r(C_3,G)\leq 2q+1$ for every isolated-free graph G with
$q$ edges was proved independently by Sidorenko and Goddard and Klietman. In
this paper instead of $C_3$ we consider $K_{2,k}$ and seek a sharp upper bound
for $r(K_{2,k},G)$ over all graphs $G$ with $q$ edges. More specifically if
$q\geq 2$, we will show that $r(C_4,G)\leq kq+1$ and that equality holds if $G
\cong qK_2$ or $K_3$. Using this we will generalize this result for
$r(K_{2,k},G)$ when $k>2$. We will also show that for every graph $G$ with $q
\geq 2$ edges and with no isolated vertices, $r(C_4, G) \leq 2p+ q - 2$ where
$p=|V(G)|$ and that equality holds if $G \cong K_3$.
