2008/10/22 by Gábor Elek, Elek, Gábor, Balázs Szegedy +1 · 4 citations
Mathematics · #05C99 #82B99 #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.0810.4062
openalex publication_date 2008/10/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we develop a measure-theoretic method to treat problems in hypergraph theory. Our central theorem is a correspondence principle between three objects: An increasing hypergraph sequence, a measurable set in an ultraproduct space and a measurable set in a finite dimensional Lebesgue space. Using this correspondence principle we build up the theory of dense hypergraphs from scratch. Along these lines we give new proofs for the Hypergraph Removal Lemma, the Hypergraph Regularity Lemma, the Counting Lemma and the Testability of Hereditary Hypergraph Properties. We prove various new results including a strengthening of the Regularity Lemma and an Inverse Counting Lemma. We also prove the equivalence of various notions for convergence of hypergraphs and we construct limit objects for such sequences. We prove that the limit objects are unique up to a certain family of measure preserving transformations. As our main tool we study the integral and measure theory on the ultraproduct of finite measure spaces which is interesting on its own right.