Size of order-preserving minimal perfect hash family
Reference [1] defines order-preserving as follows:


  A hash function is order preserving if it puts entries into the hash table in a prespecified order.


In your case, this means that given any three elements $x,y,z \in \{1,\ldots,8\}$, you want there to be a hash function which maps $x$ to $1$, $y$ to $2$, $z$ to $3$.

In particular, it need not be the case that $x &lt; y &lt; z$.

If you add the assumption that $x &lt; y &lt; z$, then you can remove the $m!$ factor in the lower bound on $H$. The new lower bound states $H \geq 2.95$, that is, $H \geq 3$, which is not tight.