Online size Ramsey number for C4 and P6

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