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In this paper we consider a game played on the edge set of the infiniteclique $K_\mathbb{N}$ by two players, Builder and Painter. In each round of thegame, Builder chooses an edge and Painter colors it red or blue. Builder winswhen Painter creates a red copy of $G$ or a blue copy of $H$, for some fixedgraphs $G$ and $H$. Builder wants to win in as few rounds as possible, andPainter wants to delay Builder for as many rounds as possible. The online sizeRamsey number $\tilde{r}(G,H)$, is the minimum number of rounds within whichBuilder can win, assuming both players play optimally. So far it has beenproven by Dybizba\'nski, Dzido and Zakrzewska that$11\leq\tilde{r}(C_4,P_6)\leq13$ \cite{Dzido}. In this paper, we refine thisresult 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 sizeRamsey number

Online size Ramsey number for $C_4$ and $P_6$