Paper detail

A Removal Lemma for Ordered Hypergraphs

We prove a removal lemma for induced ordered hypergraphs, simultaneously generalizing Alon--Ben-Eliezer--Fischer&#39;s removal lemma for ordered graphs and the induced hypergraph removal lemma. That is, we show that if an ordered hypergraph $(V,G,<)$ has few induced copies of a small ordered hypergraph $(W,H,\prec)$ then there is a small modification $G&#39;$ so that $(V,G&#39;,<)$ has no induced copies of $(W,H,\prec)$. (Note that we do \emph{not} need to modify the ordering $<$.) We give our proof in the setting of an ultraproduct (that is, a Keisler graded probability space), where we can give an abstract formulation of hypergraph removal in terms of sequences of $σ$-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&#39;&#39; set. Along the way we give an explicit construction of the bijection between the ultraproduct limit object and the corresponding hyerpgraphon.