arXiv subjects

Clark Lyons

Publications and source records attributed to Clark Lyons.

4 recordsLinked to original sources

arXiv2025

The ineffectiveness of the regularity lemma for bounded degree graphs

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.

math.CO↗

arXiv2025

Separating complexity classes of LCL problems on grids

We study the complexity of locally checkable labeling (LCL) problems on $\mathbb{Z}^n$ from the point of view of descriptive set theory, computability theory, and factors of i.i.d. Our results separate various complexity classes that were not previously known to be distinct and serve as counterexamples to a number of natural conjectures in the field.

math.LO↗

arXiv2024

Borel Families of Games

We give an elementary proof that in a Borel family of games, the set of games for which player II has a winning strategy is Baire measurable, universally measurable, and completely Ramsey in the case where $X = [\mathbb{N}]^{\aleph_0}$.

math.LO↗

arXiv2023

Baire Measurable Matchings in Non-Amenable Graphs

We prove that every Schreier graph of a free Borel action of a finitely generated non-amenable group admits a Baire measurable perfect matching, and that the Schreier graph of a free computable action of a finitely generated non-amenable group admits a computable perfect matching. These results were previously only known in the bipartite setting. To prove them, we establish variants of Tutte's theorem on perfect matchings in non-bipartite graphs. We also prove that every Borel non-amenable bounded-degree graph with only even degrees admits a Baire measurable balanced orientation.

math.LO↗