${\le }_{SP}$ Can Have Infinitely Many Classes

Building off of recent results on Keisler’s order, we show that consistently, ${\le }_{SP}$ has infinitely many classes. In particular, we define the property of $\le k$-type amalgamation for simple theories, for each $2 \leq k < \omega$\mathrm{.}Ifwelet$T_{n, k}$bethetheoryoftherandom$k$-ary,$n$-clique free random hyper-graph, then $T_{n,k}$ has $\le k-1$-type amalgamation but not $\le k$-type amalgamation. We show that consistently, if $T$ has $\leq k$-typeamalgamationthen$T_{k+1, k} \not \leq_{SP} T$, thus producing infinitely many ${\le }_{SP}$-classes. The same construction gives a simplified proof of Shelah’s theorem that consistently, the maximal ${\le }_{SP}$-class is exactly the class of unsimple theories. Finally, we show that consistently, if $T$ has $<{\mathrm{\aleph }}_0$-type amalgamation, then $T{\le }_{SP}{T}_{rg}$, the theory of the random graph.