TOWARDS A REGULARITY LEMMA FOR HIGHER ARITY DISTAL STRUCTURES

Abstract

Much recent work in model theory has investigated certain dividing lines in the class of first-order theories. Dividing lines are intended to constitute boundaries between theories that admit a certain kind of analysis on one side and comparatively much more complicated theories on the other. More recently work has been done to generalize many of these properties to a higher arity setting — classically definitions have been made with reference to the properties of formulas with a fixed binary partition of the variables. We study structures satisfying higher arity generalizations of distality and NIP. A different thread that we pick up is more combinatorial in nature. In the 1970s, Szemerédi proved his noted Regularity Lemma for graphs. Since then, much work has been devoted to proving stronger analogues of this result for restricted classes of graphs. We prove results in the direction of a regularity lemma for higher arity distal structures. Our main result establishes that under the assumptions of NIP and strong n-distality extensions of certain n-fold tensor-products of measures over small models are determined by the extension of their (n−1)-fold restrictions, a higher arity generalisation of the notion of a smooth measure. A conse- quence of this, under the additional assumption that the theory has definable Skolem functions, is that certain (n + 1)-fold tensor products of measures are determined globally by their n-fold restrictions. There remain many avenues for extending this result, which we note in the final section.