Summary: Given two graph families \(\mathcal H_1\) and \(\mathcal H_2\), 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 is trying to force Painter to create a red copy of a graph from \(\mathcal H_1\) or a blue copy of a graph from \(\mathcal H_2\) as soon as possible. The online (size) Ramsey number \(\tilde{r}(\mathcal H_1,\mathcal H_2)\) is the smallest number of rounds in the game provided Builder and Painter play optimally. We prove that if \(\mathcal H_1\) is the family of all odd cycles and \(\mathcal H_2\) is the family of all connected graphs on \(n\) vertices and \(m\) edges, then \(\tilde{r}(\mathcal H_1,\mathcal H_2)\geqslant \varphi n + m-2\varphi+1\), where \(\varphi\) is the golden ratio, and for \(n\geqslant 3, m\le (n-1)^2/4\) we have \(\tilde{r}(\mathcal H_1,\mathcal H_2)\leqslant n+2m+O(\sqrt{m-n+1})\). We also show that \(\tilde{r}(C_3,P_n)\leqslant 3n-4\) for \(n\geqslant 3\). As a consequence we get \(2.6n-3\leqslant \tilde{r}(C_3,P_n)\leqslant 3n-4\) for every \(n\geqslant 3\).