The ineffectiveness of the regularity lemma for bounded degree graphs

TL;DR

This paper demonstrates that the regularity lemma cannot be effectively applied to bounded degree graphs for approximation purposes, providing a negative answer to a longstanding open question.

Contribution

It establishes the non-existence of a computable bound for graph size needed to approximate bounded degree graphs, refuting a question by Lovász.

Findings

No computable bound exists for approximation of bounded degree graphs

Refutes the Aldous-Lyons conjecture through recent work

Provides a negative answer to a question by Lovász

Abstract

We show that for any $\Delta \geq 3$, there is no bound computable from $(\varepsilon, r)$ on the size of a graph required to approximate a graph of maximum degree at most $\Delta$ up to $\varepsilon$ error in $r$-neighborhood statistics. This provides a negative answer to a question posed by Lov\'asz. Our result is a direct consequence of the recent celebrated work of Bowen, Chapman, Lubotzky, and Vidick, which refutes the Aldous-Lyons conjecture.