Indivisible homogeneous directed graphs and a game for vertex partitions

Abstract

Let $\mathcal{T}$ be a set of finite tournaments. We will give a necessary and sufficient condition for the $\mathcal{T}$-free homogeneous directed graph $H_{\mathcal{T}}$ to be divisible; that is, that there is a partition of $H_{\mathcal{T}}$ into two sets neither of which contains an isomorphic copy of $H_{\mathcal{T}}$.