Limits of Hypergraphs, Removal and Regularity Lemmas. A Non-standard Approach

TL;DR

This paper introduces a novel approach to hypergraph theory by utilizing ultraproducts to construct limit objects, providing new proofs for key lemmas like the Hypergraph Removal and Regularity Lemmas.

Contribution

It presents a non-standard method using ultraproducts to analyze hypergraphs, offering alternative proofs for fundamental lemmas in the field.

Findings

Constructed limit objects for hypergraph sequences.

Provided new proofs for the Hypergraph Removal Lemma.

Developed a measure-theoretic framework for hypergraph analysis.

Abstract

We study the integral and measure theory of the ultraproduct of finite sets. As a main application we construct limit objects for hypergraph sequences. We give a new proof for the Hypergraph Removal Lemma and the Hypergraph Regularity Lemma.