Online size Ramsey numbers: Path vs $C_4$

Abstract: Given two graphs $G$ and $H$, a size Ramsey game is played on the edge set of $K_\mathbb{N}$. In every round, Builder selects an edge and Painter colours it red or blue. Builder's goal is to force Painter to create a red copy of $G$ or a blue copy of $H$ as soon as possible. The online (size) Ramsey number $\tilde r(G,H)$ is the number of rounds in the game provided Builder and Painter play optimally. We prove that $\tilde r(C_4,P_n)\le 2n-2$ for every $n\ge 8$. The upper bound matches the lower bound obtained by J. Cyman, T. Dzido, J. Lapinskas, and A. Lo, so we get $\tilde r(C_4,P_n)=2n-2$ for $n\ge 8$. Our proof for $n\le 13$ is computer assisted. The bound $\tilde r(C_4,P_n)\le 2n-2$ solves also the "all cycles vs. $P_n$" game for $n\ge 8$ $-$ it implies that it takes Builder $2n-2$ rounds to force Painter to create a blue path on $n$ vertices or any red cycle.