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arXiv · 2505.06215

The ineffectiveness of the regularity lemma for bounded degree graphs

Clark Lyons·Grigory Terlov·Zoltán Vidnyánszky
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Abstract

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

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Clark Lyons, Grigory Terlov, Zoltán Vidnyánszky. 2025-08-12. The ineffectiveness of the regularity lemma for bounded degree graphs. https://arxiv.org/abs/2505.06215

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