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arXiv · 2101.09769

A Removal Lemma for Ordered Hypergraphs

Henry Towsner
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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 $(V,G,<)$ has few induced copies of a small ordered hypergraph $(W,H,\prec)$ then there is a small modification $G'$ so that $(V,G',<)$ 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 $\sigma$-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.

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Henry Towsner. 2021-01-24. A Removal Lemma for Ordered Hypergraphs. https://arxiv.org/abs/2101.09769

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