Local problems on trees from the perspectives of distributed algorithms, finitary factors, and descriptive combinatorics
We study connections between distributed local algorithms, finitary factors of iid processes,
and descriptive combinatorics in the context of regular trees. We extend the Borel …
and descriptive combinatorics in the context of regular trees. We extend the Borel …
Borel Vizing's theorem for graphs of subexponential growth
A Bernshteyn, A Dhawan - Proceedings of the American Mathematical …, 2025 - ams.org
We show that every Borel graph $ G $ of subexponential growth has a Borel proper edge-
coloring with $\Delta (G)+ 1$ colors. We deduce this from a stronger result, namely that an …
coloring with $\Delta (G)+ 1$ colors. We deduce this from a stronger result, namely that an …
The landscape of distributed complexities on trees and beyond
We study the local complexity landscape of locally checkable labeling (LCL) problems on
constant-degree graphs with a focus on complexities below log* n. Our contribution is …
constant-degree graphs with a focus on complexities below log* n. Our contribution is …
Borel versions of the Local Lemma and LOCAL algorithms for graphs of finite asymptotic separation index
A Bernshteyn, F Weilacher - arXiv preprint arXiv:2308.14941, 2023 - arxiv.org
Asymptotic separation index is a parameter that measures how easily a Borel graph can be
approximated by its subgraphs with finite components. In contrast to the more classical …
approximated by its subgraphs with finite components. In contrast to the more classical …
Borel edge colorings for finite-dimensional groups
F Weilacher - Israel Journal of Mathematics, 2024 - Springer
We study the potential of Borel asymptotic dimension, a tool introduced recently in [2], to
help produce Borel edge colorings of Schreier graphs generated by Borel group actions. We …
help produce Borel edge colorings of Schreier graphs generated by Borel group actions. We …
Descriptive combinatorics, computable combinatorics, and ASI algorithms
L Qian, F Weilacher - arXiv preprint arXiv:2206.08426, 2022 - arxiv.org
We introduce new types of local algorithms, which we call" ASI Algorithms", and use them to
demonstrate a link between descriptive and computable combinatorics. This allows us to …
demonstrate a link between descriptive and computable combinatorics. This allows us to …
Circle squaring with pieces of small boundary and low Borel complexity
Tarski's Circle Squaring Problem from 1925 asks whether it is possible to partition a disk in
the plane into finitely many pieces and reassemble them via isometries to yield a partition of …
the plane into finitely many pieces and reassemble them via isometries to yield a partition of …
The Distributed Complexity of Locally Checkable Labeling Problems Beyond Paths and Trees
YJ Chang - arXiv preprint arXiv:2311.06726, 2023 - arxiv.org
We consider locally checkable labeling LCL problems in the LOCAL model of distributed
computing. Since 2016, there has been a substantial body of work examining the possible …
computing. Since 2016, there has been a substantial body of work examining the possible …
Borel combinatorics of abelian group actions
S Gao, S Jackson, E Krohne, B Seward - arXiv preprint arXiv:2401.13866, 2024 - arxiv.org
We study the free part of the Bernoulli action of $\mathbb {Z}^ n $ for $ n\geq 2$ and the
Borel combinatorics of the associated Schreier graphs. We construct orthogonal …
Borel combinatorics of the associated Schreier graphs. We construct orthogonal …