  Let $U_5$ be the tournament with vertices $v_1$, ..., $v_5$ such that $v_2
\rightarrow v_1$, and $v_i \rightarrow v_j$ if $j-i \equiv 1$, $2 \pmod{5}$ and
${i,j} \neq {1,2}$. In this paper we describe the tournaments which do not have
$U_5$ as a subtournament. Specifically, we show that if a tournament $G$ is
"prime"---that is, if there is no subset $X \subseteq V(G)$, $1 < |X| <
|V(G)|$, such that for all $v \in V(G) \backslash X$, either $v \rightarrow x$
for all $x \in X$ or $x \rightarrow v$ for all $x \in X$---then $G$ is
$U_5$-free if and only if either $G$ is a specific tournament $T_n$ or $V(G)$
can be partitioned into sets $X$, $Y$, $Z$ such that $X \cup Y$, $Y \cup Z$,
and $Z \cup X$ are transitive. From the prime $U_5$-free tournaments we can
construct all the $U_5$-free tournaments. We use the theorem to show that every
$U_5$-free tournament with $n$ vertices has a transitive subtournament with at
least $n^{\log_3 2}$ vertices, and that this bound is tight.
