Recursive upper bounds for the vertex online Ramsey game with applications to hypergraph Ramsey numbers

The classical recursive upper bound on hypergraph Ramsey numbers due to Erdős and Rado states that for $2\le k<s\le t$, $$r_k(s,t)\le 2^{\left(\genfrac{}{}{0px}{}{r_{k-1}(s-1,t-1)}{k-1}\right)}\mathrm{.}$$ In 2010, Conlon, Fox, and Sudakov introduced the so-called vertex online Ramsey numbers $\tilde{r}(s,t)$ for graphs to obtain a quantitative improvement over this bound when $k=3$. In this note, we show that the natural hypergraph generalization ${\tilde{r}}_k(s,t)$ of the vertex online Ramsey numbers satisfy an improved recurrence $${\tilde{r}}_k(s,t)\le 2^{(1+o(1)){\tilde{r}}_{k-1}(s-1,t-1)}\mathrm{.}$$ We obtain several corollaries from this, including a lower-order improvement to the best known quantitative upper bounds for hypergraph Ramsey numbers and an improvement to the above recursive bound of Erdős and Rado.