  Building off of recent results on Keisler's order, we show that consistently,
$\leq_{SP}$ has infinitely many classes. In particular, we define the property
of $\leq k$-type amalgamation for simple theories, for each $2 \leq k <
\omega$. If we let $T_{n, k}$ be the theory of the random $k$-ary, $n$-clique
free random hyper-graph, then $T_{n, k}$ has $\leq k-1$-type amalgamation but
not $\leq k$-type amalgamation. We show that consistently, if $T$ has $\leq
k$-type amalgamation then $T_{k+1, k} \not \leq_{SP} T$, thus producing
infinitely many $\leq_{SP}$-classes. The same construction gives a simplified
proof of Shelah's theorem that consistently, the maximal $\leq_{SP}$-class is
exactly the class of unsimple theories. Finally, we show that consistently, if
$T$ has $<\aleph_0$-type amalgamation, then $T \leq_{SP} T_{rg}$, the theory of
the random graph.
