The number of 3-SAT functions

With $G_k(n)$ the number of functions of $n$ boolean variables definable by $k$-SAT formulae, we prove that $G_3(n)$ is asymptotic to $2^{n+\binom{n}{3}}$. This is a strong form of the case $k=3$ of a conjecture of Bollobás, Brightwell and Leader stating that for fixed $k$, $\log_2 G_k(n)\sim \binom{n}{k}$.

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