  In this paper we consider a game played on the edge set of the infinite
clique $K_\mathbb{N}$ by two players, Builder and Painter. In each round of the
game, Builder chooses an edge and Painter colors it red or blue. Builder wins
when Painter creates a red copy of $G$ or a blue copy of $H$, for some fixed
graphs $G$ and $H$. Builder wants to win in as few rounds as possible, and
Painter wants to delay Builder for as many rounds as possible. The online size
Ramsey number $\tilde{r}(G,H)$, is the minimum number of rounds within which
Builder can win, assuming both players play optimally. So far it has been
proven by Dybizba\'nski, Dzido and Zakrzewska that
$11\leq\tilde{r}(C_4,P_6)\leq13$ \cite{Dzido}. In this paper, we refine this
result and show the exact value, namely we will present the Theorem that
$\tilde{r}(C_4,P_6)=11$, with the details of the proof.
  Keywords: graph theory, Ramsey theory, combinatorial games, online size
Ramsey number
