On 1-subdivisions of transitive tournaments

The oriented Ramsey number $\vec{r}(H)$ for an acyclic digraph $H$ is the minimum integer $n$ such that any $n$-vertex tournament contains a copy of $H$ as a subgraph. We prove that the $1$-subdivision of the $k$-vertex transitive tournament $H_k$ satisfies $\vec{r}(H_k)=O(k^2\log \log k)$. This is tight up to multiplicative $\log \log k$-term. We also show that if $T$ is an $n$-vertex tournament with ${\mathrm{\Delta }}^{+}(T)-{\delta }^{+}(T)=O(n\mathrm{/}k)-k^2$, then $T$ contains a $1$-subdivision of ${\vec{K}}_k$, a complete $k$-vertex digraph with all possible $k(k-1)$ arcs. This is also tight up to multiplicative constant.