Local Ihara’s Lemma and Applications

A Sharper Local Lemma with Improved Applications

We give a new family of Lovász Local Lemmas (LLL), with applications. Shearer has given the most general condition under which the LLL holds, but the original condition of Lovász is simpler and more practical. Do we have to make a choice between practical and optimal? In this article we present a continuum of LLLs between the original and Shearer’s conditions. One of these, which we call Clique...

Constructive Lovasz Local Lemma

Let V be a finite set of independent random variables, and let A denote a finite set of events that are determined by V . That is, each event A ∈ A maps the set of assignments of V to {0, 1}. Definition 1. Given the set of independent random variables V and set of events A determined by the variables of V , define the relevant variables for an event A ∈ A, denoted vbl(A) ⊂ V to be the smallest ...

Lovász Local Lemma

Zorn’s Lemma and Some Applications

Zorn’s lemma is a result in set theory that appears in proofs of some non-constructive existence theorems throughout mathematics. We will state Zorn’s lemma below and use it in later sections to prove some results in linear algebra, ring theory, and group theory. In an appendix, we will give an application to metric spaces. The statement of Zorn’s lemma is not intuitive, and some of the termino...

Hensel’s Lemma and Some Applications

The purpose of this note is to record Hensel’s Lemma and a few of its immediate applications for my Math 831 class. At the end of the 19th century Hensel introduced the notion of completion into a number theoretic context in order to bring to bear techniques of analysis on some purely algebraic problems in number theory. One of his most crucial discoveries (in modern terms) was that when a ring...