The ACO Seminar (2022–2023)

Given an $n$-vertex oriented tree $T$, what is the smallest size a tournament $G$ must be in order to guarantee $G$ contains a copy of $T$? A strengthening of Sumner’s conjecture poses that it is enough for $G$ to have $(n+k-1)$ vertices, where $k$ is the number of leaves of $T$. In this talk we will look at recent progress towards this conjecture. We shall also consider how this problem can be addressed by instead considering the maximum degree of the tree, rather than the number of leaves, and state some open problems in this maximum degree setting. This is joint work with Richard Montgomery.