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Borel version of the Local Lemma

Endre Csóka, Łukasz Grabowski, András Máthé, Oleg Pikhurko, Konstantinos Tyros

Abstract

We prove a Borel version of the local lemma, i.e. we show that, under suitable assumptions, if the set of variables in the local lemma has a structure of a Borel space, then there exists a satisfying assignment which is a Borel function. The main tool which we develop for the proof, which is of independent interest, is a parallel version of the Moser-Tardos algorithm which uses the same random bits to resample clauses that are far enough in the dependency graph.

Borel version of the Local Lemma

Abstract

We prove a Borel version of the local lemma, i.e. we show that, under suitable assumptions, if the set of variables in the local lemma has a structure of a Borel space, then there exists a satisfying assignment which is a Borel function. The main tool which we develop for the proof, which is of independent interest, is a parallel version of the Moser-Tardos algorithm which uses the same random bits to resample clauses that are far enough in the dependency graph.

Paper Structure

This paper contains 14 sections, 23 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Basic notation and definitions
  3. Moser-Tardos algorithm
  4. Borel graphs
  5. Moser-Tardos algorithm with limited randomness
  6. Preliminaries on landscapes of trees
  7. Landscapes
  8. Grounded landscapes
  9. Counting grounded landscapes
  10. The $b$-valued sequences encoded by a decorated landscape
  11. Landscape restrictions
  12. Local Lemma on Borel graphs of subexponential growth
  13. Analysis of the Moser-Tardos algorithm
  14. Borel Local Lemma

Key Result

Theorem 1.1

If for all $x\in V(G)$ we have $p(x) < \frac{1}{e{\Delta}}$ then there exists $f\colon V(G)\to b$ which satisfies $\mathbf R$.

Theorems & Definitions (50)

  • Theorem 1.1: Lovász's Local Lemma MR0491337
  • Corollary 1.2: MR0382050
  • proof
  • Theorem 1.3: Borel Local Lemma
  • Corollary 1.4
  • Definition 1.6
  • Theorem 1.7: Measurable Local Lemma
  • Definition 1.8
  • Example 1.9
  • Lemma 2.1
  • ...and 40 more