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 NFOPₖ, 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 NIP theory with unbounded k-splitting for every k. Finally, as a thematically related but technically independent result, we show that treelessness implies NFOP₂, sharpening a result of Kaplan, Ramsey, and Simon.