On 1-subdivisions of transitive tournaments

TL;DR

This paper investigates the oriented Ramsey number for 1-subdivisions of transitive tournaments, establishing tight bounds and conditions for their presence in large tournaments.

Contribution

It provides tight bounds on the oriented Ramsey number for 1-subdivisions of transitive tournaments and characterizes degree conditions for containing such subdivisions.

Findings

Bound $oldsymbol{ angle}$ for 1-subdivisions of transitive tournaments as $O(k^2 ext{loglog}k)$.

Characterized degree imbalance conditions ensuring the existence of 1-subdivisions of complete digraphs.

Results are tight up to logarithmic factors.

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.