Why is Lovasz Local Lemma "local" and question about its versions

Question

I have some questions about Lovasz's Local Lemma. There are two versions:

1. (symmetric version):symmetric Lovasz Local Lemma

2. (Asymmetric version):asymmetric Lovasz Local Lemma.

My questions are:

1.What is symmetric about the symmeteric version and what is asymmetric about the asymmetric version?

2.Why is the Lemma called "Local"?

Thanks for your help.

Answer 1

What is symmetric about the symmetric version and what is asymmetric about the asymmetric version?

In the asymmetric version, there is a function $x\colon\mathcal{A}\to [0,1)$: its value depends on the event $A$ considered, and is not necessarily the same for all events.

In the symmetric version, that $x$ is constant: $x(A)=1/(d+1)$ for every event $A$ considered. That's symmetric (all events are "treated" the same.)

"such that each event is independent of all the other events except for at most d of them."

You only have local dependencies: no event depends globally on every other, only on a few.