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In this paper we investigate actions of $\operatorname{SAut}(F_n)$, theunique index $2$ subgroup of $\operatorname{Aut}(F_n)$, on small sets,improving upon results by Baumeister--Kielak--Pierro for several small valuesof $n$. Using a computational approach for $n \geqslant 5$, we show that everyaction of $\operatorname{SAut}(F_n)$ on a set containing fewer than $18$elements is trivial.

On the Smallest Non-trivial Action of SAut(Fn) for Small n