The online Ramsey number conjecture for cycles of fixed odd length

Let $\tilde{r}(C_k,C_n)$ denote the online Ramsey number for forcing a red cycle $C_k$ or a blue cycle $C_n$. Here $k$ is fixed and odd, with $k\ge 3$, while $n$ tends to infinity.

Online cycle Ramsey conjecture.

$$\tilde{r}(C_k,C_n)=3n+o(n).$$

The paper proves the asymptotically tight value $2n+o(n)$ when $k$ is fixed and even, while the corresponding result for odd $k$ is not known. Lower bounds and the methods developed in the paper suggest that the odd-cycle case is asymptotic to $3n$.