Directed Ramsey number for trees

Abstract

In this paper, we study Ramsey-type problems for directed graphs. We first consider the $k$-colour oriented Ramsey number of $H$, denoted by $\overrightarrow{R}(H,k)$, which is the least $n$ for which every $k$-edge-coloured tournament on $n$ vertices contains a monochromatic copy of $H$. We prove that $\overrightarrow{R}(T,k)\le c_k|T{|}^k$ for any oriented tree $T$. This is a generalisation of a similar result for directed paths by Chvátal and by Gyárfás and Lehel, and answers a question of Yuster. In general, it is tight up to a constant factor. We also consider the $k$-colour directed Ramsey number $\overleftrightarrow{R}(H,k)$ of $H$, which is defined as above, but, instead of colouring tournaments, we colour the complete directed graph of order $n$. Here we show that $\overleftrightarrow{R}(T,k)\le c_k|T{|}^{k-1}$ for any oriented tree $T$, which is again tight up to a constant factor, and it generalises a result by Williamson and by Gyárfás and Lehel who determined the $2$-colour directed Ramsey number of directed paths.

27 pages, 14 figures

References in corpus (1)

• Monochromatic paths in random tournaments

Cited by in corpus (1)

• Monochromatic trees in random tournaments