The Local Cut Lemma

TL;DR

The paper introduces the Local Cut Lemma, a probabilistic tool that generalizes the Lovász Local Lemma, providing explicit bounds and new combinatorial applications without relying on entropy compression.

Contribution

It presents the Local Cut Lemma, a new probabilistic lemma that extends the Lovász Local Lemma and simplifies analysis with explicit probability bounds.

Findings

Provides a probabilistic proof of the Local Cut Lemma

Yields explicit lower bounds for probabilities of certain events

Applies to improve bounds in color-critical hypergraphs

Abstract

The Lov\'{a}sz Local Lemma is a very powerful tool in probabilistic combinatorics, that is often used to prove existence of combinatorial objects satisfying certain constraints. Moser and Tardos have shown that the LLL gives more than just pure existence results: there is an effective randomized algorithm that can be used to find a desired object. In order to analyze this algorithm, Moser and Tardos developed the so-called entropy compression method. It turned out that one could obtain better combinatorial results by a direct application of the entropy compression method rather than simply appealing to the LLL. The aim of this paper is to provide a generalization of the LLL which implies these new combinatorial results. This generalization, which we call the Local Cut Lemma, concerns a random cut in a directed graph with certain properties. Note that our result has a short probabilistic proof that does not use entropy compression. As a consequence, it not only shows that a certain probability is positive, but also gives an explicit lower bound for this probability. As an illustration, we present a new application (an improved lower bound on the number of edges in color-critical hypergraphs) as well as explain how to use the Local Cut Lemma to derive some of the results obtained previously using the entropy compression method.