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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k-triviality collapses to triviality among simple theories, enabling new non-k-ary examples of k-distal theories and showing distality properties fail to be preserved under reducts.
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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 triviality among simple theories, enabling new non-k-ary examples of k-distal theories and showing distality properties fail to be preserved under reducts.