Hypergraph limits: a regularity approach

TL;DR

This paper introduces a new proof and construction for hypergraph limits using a regularity approach inspired by Lovász and Szegedy, providing a more accessible framework for understanding hypergraph convergence.

Contribution

It offers a novel proof and construction of hypergraph limits based on a weak regularity lemma, simplifying previous ultraproduct-based methods.

Findings

Provides a new, more accessible proof of hypergraph limits

Constructs hypergraph limits using a regularity approach

Connects hypergraph limits with classical regularity lemmas

Abstract

A sequence of $k$-uniform hypergraphs $H_1, H_2, \dots$ is convergent if the sequence of homomorphism densities $t(F, H_1), t(F, H_2), \dots$ converges for every $k$-uniform hypergraph $F$. For graphs, Lov\'asz and Szegedy showed that every convergent sequence has a limit in the form of a symmetric measurable function $W \colon [0,1]^2 \to [0,1]$. For hypergraphs, analogous limits $W \colon [0,1]^{2^k-2} \to [0,1]$ were constructed by Elek and Szegedy using ultraproducts. These limits had also been studied earlier by Hoover, Aldous, and Kallenberg in the setting of exchangeable random arrays. In this paper, we give a new proof and construction of hypergraph limits. Our approach is inspired by the original approach of Lov\'asz and Szegedy, with the key ingredient being a weak Frieze-Kannan type regularity lemma.