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 tight up to multiplicative constant.
@article{10_37236_10788,
author = {Jaehoon Kim and Hyunwoo Lee and Jaehyeon Seo},
title = {On 1-subdivisions of transitive tournaments},
journal = {The electronic journal of combinatorics},
year = {2022},
volume = {29},
number = {1},
doi = {10.37236/10788},
zbl = {1486.05118},
url = {http://geodesic.mathdoc.fr/articles/10.37236/10788/}
}
TY - JOUR
AU - Jaehoon Kim
AU - Hyunwoo Lee
AU - Jaehyeon Seo
TI - On 1-subdivisions of transitive tournaments
JO - The electronic journal of combinatorics
PY - 2022
VL - 29
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/10788/
DO - 10.37236/10788
ID - 10_37236_10788
ER -
%0 Journal Article
%A Jaehoon Kim
%A Hyunwoo Lee
%A Jaehyeon Seo
%T On 1-subdivisions of transitive tournaments
%J The electronic journal of combinatorics
%D 2022
%V 29
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/10788/
%R 10.37236/10788
%F 10_37236_10788
Jaehoon Kim; Hyunwoo Lee; Jaehyeon Seo. On 1-subdivisions of transitive tournaments. The electronic journal of combinatorics, Tome 29 (2022) no. 1. doi: 10.37236/10788
