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 Erd\H{o}s and Rado states that for $2 \leq k < s \leq t$, $$r_k(s,t) \leq 2^{\binom{r_{k-1}(s-1,t-1)}{k-1}}.$$ 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 $$\tilde{r}_k(s,t) \leq 2^{(1+o(1))\tilde{r}_{k-1}(s-1,t-1)}.$$ 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 Erd\H{o}s and Rado.