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arXiv · 2110.05002

On 1-subdivisions of transitive tournaments

Jaehoon Kim·Hyunwoo Lee·Jaehyeon Seo
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Abstract

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 $\Delta^+(T)-\delta^+(T)= O(n/k) - k^2$, 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.

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Jaehoon Kim, Hyunwoo Lee, Jaehyeon Seo. 2021-10-11. On 1-subdivisions of transitive tournaments. https://doi.org/10.37236/10788

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