A removal lemma for ordered hypergraphs

Abstract We prove a removal lemma for induced ordered hypergraphs, simultaneously generalizing Alon–Ben‐Eliezer–Fischer's removal lemma for ordered graphs and the induced hypergraph removal lemma. That is, we show that if an ordered hypergraph has few induced copies of a small ordered hypergraph then there is a small modification so that has no induced copies of . (Note that we do not need to modify the ordering .) We give our proof in the setting of an ultraproduct (i.e., a Keisler graded probability space), where we can give an abstract formulation of hypergraph removal in terms of sequences of ‐algebras. We then show that ordered hypergraphs can be viewed as hypergraphs where we view the intervals as an additional notion of a “very structured” set. Along the way we give an explicit construction of the bijection between the ultraproduct limit object and the corresponding hyerpgraphon.