  Given two graphs $G$ and $H$ with $H\subseteq G$ we consider the anti-Ramsey
function $AR(G,H)$ which is the maximum number of colors in any edge-coloring
of $G$ so that every copy of $H$ receives the same color on at least one pair
of edges. The classical Tur\'an function for a graph $G$ and family of graphs
$\mathcal{F}$, written $ex(G,\mathcal{F})$, is defined as the maximum number of
edges of a subgraph of $G$ not containing any member of $\mathcal{F}$. We show
that there exists a constant $c>0$ so that $AR(K_n,K_{s,t})-ex(K_n,K_{s,t})<cn$
and $c$ depends only on $s$ and $t$, which implies $AR(K_n,K_{s,t})\leq
cn^{2-\frac{1}{s}}$, for $s\leq t$ by a result of K\H ovari, S\'os, and
Tur\'an.
