The Stanley–Wilf conjecture for hereditary properties of tournaments

Let $cmathcal{P}$ be a hereditary property of tournaments, and let $cmathcal{P}_n$ denote the tournaments in $cmathcal{P}$ on $n$ vertices. There exists a constant $calpha>0$ and a function $F(n)=n^{calpha n+o(n)}$ such that, for every hereditary property of tournaments $cmathcal{P}$, either

$$|\mathcal{P}_n|\leq c^n$$

for every $n\in\mathbb{N}$ and some constant $c=c(\mathcal{P})$, or

$$|\mathcal{P}_n|\geq F(n)$$

for every $n\in\mathbb{N}$. This conjectures a jump from exponential to factorial speed; the paper presents it as a tournament analogue of the Stanley–Wilf conjecture, and the general assertion remains open.