  We consider asymptotics of set partition pattern avoidance in the sense of
Klazar. One of the results of this paper extends work of Alweiss, and finds a
classification for set partitions $\pi$ such that the number of set partitions
of $[n]$ avoiding $\pi$ grows more slowly than $n^{cn}$ for all $c>0$. Several
conjectures are proposed, and the related question of asymptotics of parallel
($k$-tuple) permutation pattern avoidance is considered and solved completely
to within an exponential factor, generalizing Marcus and Tardos's 2004 proof of
the Stanley-Wilf Conjecture.
