A Removal Lemma for Ordered Hypergraphs

TL;DR

This paper establishes a removal lemma for induced ordered hypergraphs, extending previous results for graphs and hypergraphs, using ultraproducts and hypergraphons to analyze structural modifications needed to eliminate small induced subhypergraphs.

Contribution

It introduces a removal lemma for ordered hypergraphs that generalizes prior lemmas, employing ultraproducts and hypergraphons for a novel proof approach.

Findings

Proves a removal lemma for induced ordered hypergraphs.

Provides an explicit construction of the ultraproduct limit and hypergraphon.

Shows ordered hypergraphs can be viewed as hypergraphs with structured intervals.

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.