Strongly n-distal NIP theories admit a hypergraph regularity lemma, compact domination for definable fsg groups, and the n-distality hierarchy is strict among stable theories; infinite such fields have characteristic zero.
Irregular triads in 3-uniform hypergraphs.Memoirs of the American Mathematical Society, accepted (arXiv:2111.01737)
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k-triviality collapses to 1-triviality among simple theories, yielding new non-k-ary examples of strongly k-distal theories and implying that certain stable theories are trivial.
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On n-distality, n-triviality and hypergraph regularity in NIP theories
Strongly n-distal NIP theories admit a hypergraph regularity lemma, compact domination for definable fsg groups, and the n-distality hierarchy is strict among stable theories; infinite such fields have characteristic zero.
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Higher-arity distality and forking triviality
k-triviality collapses to 1-triviality among simple theories, yielding new non-k-ary examples of strongly k-distal theories and implying that certain stable theories are trivial.