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{\bf J\'ozsef Balogh, B\'ela Bollob\'as and Robert Morris}
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{\bf Hereditary Properties of Tournaments}
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A collection of unlabelled tournaments ${\cal P}$ is called a
\emph{hereditary property} if it is closed under isomorphism and under
taking induced sub-tournaments. The speed of ${\cal P}$ is the
function $n \mapsto |{\cal P}_n|$, where ${\cal P}_n = \{T \in {\cal
P} : |V(T)| = n\}$. In this paper, we prove that there is a jump in
the possible speeds of a hereditary property of tournaments, from
polynomial to exponential speed. Moreover, we determine the minimal
exponential speed, $|{\cal P}_n| = c^{(1+o(1))n}$, where $c \simeq
1.47$ is the largest real root of the polynomial $x^3 = x^2 + 1$, and
the unique hereditary property with this speed.
\bye