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An Elementary Proof of the Lovász Local Lemma Without Conditional Probabilities

Igal Sason

Abstract

The Lovász Local Lemma is a powerful combinatorial tool via the probabilistic method, providing a criterion under which a collection of undesirable events with limited dependencies can be avoided simultaneously with positive probability. Standard presentations of the Lovász Local Lemma typically use conditional probabilities in intermediate steps. In this letter, we present a proof that avoids conditional probabilities altogether and instead works with unconditional probability inequalities. This formulation yields a fully self-contained argument in which every step is valid without requiring the positivity of intermediate conditioning events. The resulting proof is elementary and provides a transparent presentation of the Lovász Local Lemma.

An Elementary Proof of the Lovász Local Lemma Without Conditional Probabilities

Abstract

The Lovász Local Lemma is a powerful combinatorial tool via the probabilistic method, providing a criterion under which a collection of undesirable events with limited dependencies can be avoided simultaneously with positive probability. Standard presentations of the Lovász Local Lemma typically use conditional probabilities in intermediate steps. In this letter, we present a proof that avoids conditional probabilities altogether and instead works with unconditional probability inequalities. This formulation yields a fully self-contained argument in which every step is valid without requiring the positivity of intermediate conditioning events. The resulting proof is elementary and provides a transparent presentation of the Lovász Local Lemma.
Paper Structure (2 sections, 3 theorems, 31 equations)

This paper contains 2 sections, 3 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. The Lovász Local Lemma and Its New Proof without Conditional Probabilities

Key Result

Theorem 1

Let $A_1, \ldots, A_n$ be events in an arbitrary probability space $(\Omega,\mathcal{F},\mathbb{P})$, and let $D = ([n], E)$ be a dependency digraph for the events $\{A_i\}_{i=1}^n$. Suppose that there exist $x_1, \ldots, x_n \in [0,1)$ such that Then, In particular, none of the events occurs with positive probability.

Theorems & Definitions (11)

  • Definition 1: Mutual independence
  • Definition 2: Dependency digraph
  • Remark 1
  • Remark 2: Non-uniqueness of the dependency digraph
  • Theorem 1: The Lovász Local Lemma
  • proof
  • Lemma 1
  • Remark 3: On avoiding circular reasoning
  • Theorem 2: The Lovász Local Lemma: Symmetric Case
  • proof
  • ...and 1 more