Indiscernible extraction at small large cardinals from a higher-arity stability notion

We introduce a higher-arity stability notion defined in terms of $k$-splitting, a higher-arity generalization of splitting. We show that theories with bounded $k$-splitting have improved indiscernible extraction at $k$-ineffable cardinals, and we give a non-trivial example of a theory with bounded $k$-splitting but unbounded $(k-1)$-splitting for each odd $k>1$. We also show that bounded $k$-splitting implies ${\mathrm{N}\mathrm{F}\mathrm{O}\mathrm{P}}_k$, a higher arity stability notion introduced by Terry and Wolf. We then use our indiscernible extraction result together with a construction of Kaplan and Shelah to give a strong counterexample to the converse: an $\mathrm{N}\mathrm{I}\mathrm{P}$ theory with unbounded $k$-splitting for every $k$. Finally, as a thematically related but technically independent result, we show that treelessness implies ${\mathrm{N}\mathrm{F}\mathrm{O}\mathrm{P}}_2$, sharpening a result of Kaplan, Ramsey, and Simon.