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Analysis shows no computable bound for approximating graphs with maximum degree, indicating limits of the regularity lemma.
We show that for any , there is no bound computable from on the size of a graph required to approximate a graph of maximum degree at most up to error in ‐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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Lyons et al. (2025) studied this question.
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