12 problems

Let $P_k$ denote the path on $k$ vertices, and let $\tilde{r}(P_k,P_\ell)$ be the two-color online Ramsey number for the pair of paths $P_k$ and $P_\ell$. Cyman–Dzido–Lapinskas–Lo…

Let $P_k$ denote the path on $k$ vertices, and let $\tilde{r}(P_k,P_{\ell+1})$ be the two-color online Ramsey number for the pair of paths $P_k$ and $P_{\ell+1}$. Asymptotic path o…

For each fixed $k$, define … where $P_n$ is the path on $n$ vertices and $\tilde{r}(P_k,P_n)$ is the online size Ramsey number. The path constants conjecture. … These values refine…

Let $K_t$ be the clique on $t$ vertices. The online-to-size Ramsey conjecture. … Here $\tilde{r}(G,H)$ is the online size Ramsey number and $\hat{r}(G,H)$ is the size Ramsey number…

Online cycle Ramsey conjecture.

For graphs $G$ and $H$, let $\tilde{r}(G,H)$ denote the online Ramsey number, and let $K_{1,3}$ be the star with three edges and $P_{\ell}$ the path with $\ell$ vertices. Latip–Tan…

Even-cycle online Ramsey conjecture.

Conjecture on the triangle–path online size Ramsey number.

Grytczuk–Hałuszczak–Kierstead conjecture. Builder wins the online Ramsey game for $H$ on planar graphs if and only if $H$ is an outerplanar graph.

Fix integers $m,n\ge 3$, and let $N$ vary. Monotonicity conjecture. The restricted online Ramsey number $\tilde{r}(m,n;N)$ decreases substantially as $N$ varies between the minimum…

Let $K_N$ be the complete graph on $N$ vertices. An edge is forced when Painter must color it blue because coloring it red would create a red triangle. Forced-edge conjecture. Rega…

Let $r(3,n)$ be the classical Ramsey number, and let $\tilde{r}(3,n;N)$ denote the restricted online Ramsey number on $N$ vertices. Off-diagonal restricted online Ramsey conjecture…