  Scott and Seymour conjectured the existence of a function $f \colon
\mathbb{N} \to \mathbb{N}$ such that, for every graph $G$ and tournament $T$ on
the same vertex set, $\chi(G) \geqslant f(k)$ implies that $\chi(G[N_T^+(v)])
\geqslant k$ for some vertex $v$. In this note we disprove this conjecture even
if $v$ is replaced by a vertex set of size $\mathcal{O}(\log{\lvert
V(G)\rvert})$. As a consequence, we answer in the negative a question of
Harutyunyan, Le, Thomass\'{e}, and Wu concerning the corresponding statement
where the graph $G$ is replaced by another tournament, and disprove a related
conjecture of Nguyen, Scott, and Seymour. We also show that the setting where
chromatic number is replaced by degeneracy exhibits a quite different
behaviour.
