Measurable versions of the Lov\'{a}sz Local Lemma and measurable graph colorings
Anton Bernshteyn

TL;DR
This paper extends the Lovász Local Lemma to measurable settings, providing conditions for Borel or measure-measurable functions satisfying combinatorial constraints, with applications to graph coloring and group actions.
Contribution
It introduces measurable versions of the Lovász Local Lemma applicable to Borel and invariant functions, with new bounds on measurable chromatic numbers and characterizations for amenable group actions.
Findings
Borel measurable functions can approximate local constraints within any epsilon.
Asymptotically tight bounds on the measurable chromatic number of shift-generated graphs.
Characterization of when measure-preserving actions satisfy the measurable Lovász Local Lemma.
Abstract
In this paper we investigate the extent to which the Lov\'asz Local Lemma (an important tool in probabilistic combinatorics) can be adapted for the measurable setting. In most applications, the Lov\'asz Local Lemma is used to produce a function with certain properties, where is some underlying combinatorial structure and is a (typically finite) set. Can this function be chosen to be Borel or -measurable for some probability Borel measure on (assuming that is a standard Borel space)? In the positive direction, we prove that if the set of constraints put on is, in a certain sense, "locally finite," then there is always a Borel choice for that is "-close" to satisfying these constraints, for any . Moreover, if the combinatorial structure on is "induced" by the -shift action of a countable…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAdvanced Topology and Set Theory · Limits and Structures in Graph Theory · Topological and Geometric Data Analysis
